Properties

Label 4-930e2-1.1-c1e2-0-16
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 7-s + 4·8-s + 3·9-s + 4·10-s + 11-s − 6·12-s + 4·13-s + 2·14-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s − 3·19-s + 6·20-s − 2·21-s + 2·22-s + 7·23-s − 8·24-s + 3·25-s + 8·26-s − 4·27-s + 3·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.377·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.301·11-s − 1.73·12-s + 1.10·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 0.688·19-s + 1.34·20-s − 0.436·21-s + 0.426·22-s + 1.45·23-s − 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.567727113\)
\(L(\frac12)\) \(\approx\) \(5.567727113\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T - 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 19 T + 192 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 6 T - 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 21 T + 230 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58369742668169622881392498137, −10.07896814888613568901950952352, −9.560176306576161195048511404639, −9.135189465923416730781059868944, −8.454113429484776346681778764311, −8.269220544903797731966262923998, −7.27109803947625248603649878617, −7.25349638059761408898454286542, −6.48159766284164875326425405871, −6.36442623568392223854921019561, −5.82657404658869598588534061457, −5.56865023429522812415694669481, −5.06919548701767731828346583917, −4.59322043530120736241495437638, −4.18573751729956530611062536605, −3.69405827722768577241665985395, −2.82179157383162195127400833664, −2.50779933605166344714653458320, −1.35636171358021831321444547883, −1.19219428858697637835555291853, 1.19219428858697637835555291853, 1.35636171358021831321444547883, 2.50779933605166344714653458320, 2.82179157383162195127400833664, 3.69405827722768577241665985395, 4.18573751729956530611062536605, 4.59322043530120736241495437638, 5.06919548701767731828346583917, 5.56865023429522812415694669481, 5.82657404658869598588534061457, 6.36442623568392223854921019561, 6.48159766284164875326425405871, 7.25349638059761408898454286542, 7.27109803947625248603649878617, 8.269220544903797731966262923998, 8.454113429484776346681778764311, 9.135189465923416730781059868944, 9.560176306576161195048511404639, 10.07896814888613568901950952352, 10.58369742668169622881392498137

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