Properties

Label 6-7e3-1.1-c9e3-0-0
Degree $6$
Conductor $343$
Sign $1$
Analytic cond. $46.8604$
Root an. cond. $1.89874$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 21·2-s + 84·3-s + 231·4-s + 1.55e3·5-s + 1.76e3·6-s + 7.20e3·7-s + 2.86e3·8-s − 3.89e4·9-s + 3.26e4·10-s − 3.44e3·11-s + 1.94e4·12-s − 1.97e4·13-s + 1.51e5·14-s + 1.30e5·15-s − 4.49e4·16-s + 1.01e6·17-s − 8.18e5·18-s + 2.22e5·19-s + 3.58e5·20-s + 6.05e5·21-s − 7.23e4·22-s + 1.88e6·23-s + 2.40e5·24-s − 1.85e5·25-s − 4.15e5·26-s − 3.84e6·27-s + 1.66e6·28-s + ⋯
L(s)  = 1  + 0.928·2-s + 0.598·3-s + 0.451·4-s + 1.11·5-s + 0.555·6-s + 1.13·7-s + 0.247·8-s − 1.98·9-s + 1.03·10-s − 0.0709·11-s + 0.270·12-s − 0.192·13-s + 1.05·14-s + 0.665·15-s − 0.171·16-s + 2.95·17-s − 1.83·18-s + 0.392·19-s + 0.501·20-s + 0.678·21-s − 0.0658·22-s + 1.40·23-s + 0.148·24-s − 0.0950·25-s − 0.178·26-s − 1.39·27-s + 0.511·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $1$
Analytic conductor: \(46.8604\)
Root analytic conductor: \(1.89874\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 343,\ (\ :9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(5.623412405\)
\(L(\frac12)\) \(\approx\) \(5.623412405\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ \( ( 1 - p^{4} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 21 T + 105 p T^{2} - 303 p^{3} T^{3} + 105 p^{10} T^{4} - 21 p^{18} T^{5} + p^{27} T^{6} \)
3$S_4\times C_2$ \( 1 - 28 p T + 5117 p^{2} T^{2} - 122360 p^{3} T^{3} + 5117 p^{11} T^{4} - 28 p^{19} T^{5} + p^{27} T^{6} \)
5$S_4\times C_2$ \( 1 - 1554 T + 520107 p T^{2} - 46663764 p^{2} T^{3} + 520107 p^{10} T^{4} - 1554 p^{18} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 + 3444 T + 455343105 T^{2} + 125101303155960 T^{3} + 455343105 p^{9} T^{4} + 3444 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 + 19782 T + 18882215055 T^{2} + 378008000651932 T^{3} + 18882215055 p^{9} T^{4} + 19782 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 - 1016694 T + 649258140783 T^{2} - 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} - 1016694 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 - 222852 T + 614081373717 T^{2} - 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} - 222852 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 - 81984 p T + 5417652680517 T^{2} - 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} - 81984 p^{19} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 - 4081818 T + 38739015783987 T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} - 4081818 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 - 2869440 T + 21142500166221 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} - 2869440 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 - 1395618 T + 262675972194027 T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + 262675972194027 p^{9} T^{4} - 1395618 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 + 14420658 T + 764979654799959 T^{2} + \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} + 14420658 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 + 61631172 T + 2687603003165025 T^{2} + \)\(16\!\cdots\!04\)\( p T^{3} + 2687603003165025 p^{9} T^{4} + 61631172 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 + 10368960 T + 2946826961339709 T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + 2946826961339709 p^{9} T^{4} + 10368960 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 - 67502610 T + 6295046710287531 T^{2} - \)\(20\!\cdots\!32\)\( T^{3} + 6295046710287531 p^{9} T^{4} - 67502610 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 + 42590100 T + 19076976504365997 T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} + 42590100 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 - 191746842 T + 38596678668907359 T^{2} - \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 + 255175788 T + 80172518705654361 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + 80172518705654361 p^{9} T^{4} + 255175788 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 - 296514504 T + 147895725194380437 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} - 296514504 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 - 344213310 T + 83298340302082311 T^{2} - \)\(20\!\cdots\!12\)\( T^{3} + 83298340302082311 p^{9} T^{4} - 344213310 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 960412656 T + 566786434394061357 T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} + 960412656 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 + 1100517180 T + 873896700882341301 T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + 873896700882341301 p^{9} T^{4} + 1100517180 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 - 506816478 T + 956194525794688887 T^{2} - \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} - 506816478 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 + 647498250 T + 819696244591424799 T^{2} + \)\(48\!\cdots\!84\)\( T^{3} + 819696244591424799 p^{9} T^{4} + 647498250 p^{18} T^{5} + p^{27} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.42911643506000089547874372547, −17.55766236376809216049083689603, −17.33927692521185893050578631514, −16.63157713718644056570766495857, −16.62776829643021273195390578558, −15.13086542264021621558258926276, −15.06079762327990915889930007270, −14.20223157513765993123281144803, −14.15978559133251477385959290953, −13.82330808682233201677173380978, −13.11997253337513675165158030851, −12.22064752638243949455290575120, −11.58565574284812008648425578781, −11.41703236658857597520016080649, −10.06735957078725870787461517415, −9.956179202832500850256664400361, −8.568138050088623295102875994266, −8.424105492050637531185092440215, −7.45667248779142603474364026934, −6.19743618159869923525401645917, −5.31592821540955963675027295774, −5.13978899189559338188682326891, −3.30068204782290139935686375941, −2.67225695276816042412644794101, −1.28730657737565585578833843067, 1.28730657737565585578833843067, 2.67225695276816042412644794101, 3.30068204782290139935686375941, 5.13978899189559338188682326891, 5.31592821540955963675027295774, 6.19743618159869923525401645917, 7.45667248779142603474364026934, 8.424105492050637531185092440215, 8.568138050088623295102875994266, 9.956179202832500850256664400361, 10.06735957078725870787461517415, 11.41703236658857597520016080649, 11.58565574284812008648425578781, 12.22064752638243949455290575120, 13.11997253337513675165158030851, 13.82330808682233201677173380978, 14.15978559133251477385959290953, 14.20223157513765993123281144803, 15.06079762327990915889930007270, 15.13086542264021621558258926276, 16.62776829643021273195390578558, 16.63157713718644056570766495857, 17.33927692521185893050578631514, 17.55766236376809216049083689603, 18.42911643506000089547874372547

Graph of the $Z$-function along the critical line