Properties

Label 4-21e2-1.1-c23e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $4955.15$
Root an. cond. $8.39004$
Motivic weight $23$
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  − 5.31e5·3-s − 8.38e6·4-s + 9.86e9·7-s + 1.88e11·9-s + 4.45e12·12-s + 1.49e15·19-s − 5.24e15·21-s + 1.19e16·25-s − 5.00e16·27-s − 8.27e16·28-s − 4.88e17·31-s − 1.57e18·36-s + 2.05e18·37-s + 1.01e19·43-s + 6.99e19·49-s − 7.97e20·57-s + 1.07e21·61-s + 1.85e21·63-s + 5.90e20·64-s − 1.89e21·67-s + 8.91e21·73-s − 6.33e21·75-s − 1.25e22·76-s − 2.11e20·79-s + 8.86e21·81-s + 4.39e22·84-s + 2.59e23·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 1.88·7-s + 2·9-s + 1.73·12-s + 2.95·19-s − 3.26·21-s + 25-s − 1.73·27-s − 1.88·28-s − 3.45·31-s − 2·36-s + 1.89·37-s + 1.66·43-s + 2.55·49-s − 5.11·57-s + 3.15·61-s + 3.77·63-s + 64-s − 1.89·67-s + 3.32·73-s − 1.73·75-s − 2.95·76-s − 0.0317·79-s + 81-s + 3.26·84-s + 5.98·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4955.15\)
Root analytic conductor: \(8.39004\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(1.980047470\)
\(L(\frac12)\) \(\approx\) \(1.980047470\)
\(L(\frac{25}{2})\) 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)$
bad3$C_2$ \( 1 + p^{12} T + p^{23} T^{2} \)
7$C_2$ \( 1 - 9865813063 T + p^{23} T^{2} \)
good2$C_2^2$ \( 1 + p^{23} T^{2} + p^{46} T^{4} \)
5$C_2^2$ \( 1 - p^{23} T^{2} + p^{46} T^{4} \)
11$C_2^2$ \( 1 + p^{23} T^{2} + p^{46} T^{4} \)
13$C_2$ \( ( 1 - 9849211774339 T + p^{23} T^{2} )( 1 + 9849211774339 T + p^{23} T^{2} ) \)
17$C_2^2$ \( 1 - p^{23} T^{2} + p^{46} T^{4} \)
19$C_2$ \( ( 1 - 1015192461697768 T + p^{23} T^{2} )( 1 - 484637939972981 T + p^{23} T^{2} ) \)
23$C_2^2$ \( 1 + p^{23} T^{2} + p^{46} T^{4} \)
29$C_2$ \( ( 1 - p^{23} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 234040242184219556 T + p^{23} T^{2} )( 1 + 254711111033034139 T + p^{23} T^{2} ) \)
37$C_2$ \( ( 1 - 1619710447144558417 T + p^{23} T^{2} )( 1 - 433556633400399010 T + p^{23} T^{2} ) \)
41$C_2$ \( ( 1 + p^{23} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5063598036403118305 T + p^{23} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{23} T^{2} + p^{46} T^{4} \)
53$C_2^2$ \( 1 + p^{23} T^{2} + p^{46} T^{4} \)
59$C_2^2$ \( 1 - p^{23} T^{2} + p^{46} T^{4} \)
61$C_2$ \( ( 1 - \)\(67\!\cdots\!19\)\( T + p^{23} T^{2} )( 1 - \)\(39\!\cdots\!93\)\( T + p^{23} T^{2} ) \)
67$C_2$ \( ( 1 + \)\(38\!\cdots\!48\)\( T + p^{23} T^{2} )( 1 + \)\(15\!\cdots\!45\)\( T + p^{23} T^{2} ) \)
71$C_2$ \( ( 1 - p^{23} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(52\!\cdots\!10\)\( T + p^{23} T^{2} )( 1 - \)\(37\!\cdots\!31\)\( T + p^{23} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(11\!\cdots\!03\)\( T + p^{23} T^{2} )( 1 + \)\(11\!\cdots\!16\)\( T + p^{23} T^{2} ) \)
83$C_2$ \( ( 1 + p^{23} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{23} T^{2} + p^{46} T^{4} \)
97$C_2$ \( ( 1 - \)\(87\!\cdots\!22\)\( T + p^{23} T^{2} )( 1 + \)\(87\!\cdots\!22\)\( T + p^{23} T^{2} ) \)
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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

−13.16694564848581433659405089647, −12.69384567445292265171837487965, −11.89691570558643866897033074777, −11.30177275228259221800933341925, −11.16148758290273052284797810705, −10.41672861414817540722457325835, −9.340750209581474426794812390089, −9.222832300599903201784964751967, −7.987984353548151344800905300434, −7.49224354736094772146058847012, −6.92519154502220729227694761671, −5.66014813535109076661070777147, −5.25960390992068595331298672927, −5.12364536369743120763734176788, −4.26561649546872621750489968457, −3.71990128892207607612322580791, −2.37168952741411211504883962441, −1.40314194551751284386810968994, −0.952983901044094813043617832249, −0.52437859922303641671903908166, 0.52437859922303641671903908166, 0.952983901044094813043617832249, 1.40314194551751284386810968994, 2.37168952741411211504883962441, 3.71990128892207607612322580791, 4.26561649546872621750489968457, 5.12364536369743120763734176788, 5.25960390992068595331298672927, 5.66014813535109076661070777147, 6.92519154502220729227694761671, 7.49224354736094772146058847012, 7.987984353548151344800905300434, 9.222832300599903201784964751967, 9.340750209581474426794812390089, 10.41672861414817540722457325835, 11.16148758290273052284797810705, 11.30177275228259221800933341925, 11.89691570558643866897033074777, 12.69384567445292265171837487965, 13.16694564848581433659405089647

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