Properties

Label 4-7e4-1.1-c5e2-0-2
Degree $4$
Conductor $2401$
Sign $1$
Analytic cond. $61.7608$
Root an. cond. $2.80335$
Motivic weight $5$
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  − 11·2-s + 32·4-s + 275·8-s + 243·9-s + 76·11-s − 3.02e3·16-s − 2.67e3·18-s − 836·22-s + 4.95e3·23-s + 3.12e3·25-s + 1.45e4·29-s + 8.80e3·32-s + 7.77e3·36-s + 8.88e3·37-s + 2.34e4·43-s + 2.43e3·44-s − 5.44e4·46-s − 3.43e4·50-s − 2.45e4·53-s − 1.60e5·58-s + 4.28e4·64-s − 6.93e4·67-s − 4.44e3·71-s + 6.68e4·72-s − 9.77e4·74-s − 8.01e4·79-s − 2.58e5·86-s + ⋯
L(s)  = 1  − 1.94·2-s + 4-s + 1.51·8-s + 9-s + 0.189·11-s − 2.95·16-s − 1.94·18-s − 0.368·22-s + 1.95·23-s + 25-s + 3.21·29-s + 1.51·32-s + 36-s + 1.06·37-s + 1.93·43-s + 0.189·44-s − 3.79·46-s − 1.94·50-s − 1.20·53-s − 6.25·58-s + 1.30·64-s − 1.88·67-s − 0.104·71-s + 1.51·72-s − 2.07·74-s − 1.44·79-s − 3.76·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $1$
Analytic conductor: \(61.7608\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{49} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2401,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8191008563\)
\(L(\frac12)\) \(\approx\) \(0.8191008563\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 + 11 T + 89 T^{2} + 11 p^{5} T^{3} + p^{10} T^{4} \)
3$C_2$ \( ( 1 - p^{3} T + p^{5} T^{2} )( 1 + p^{3} T + p^{5} T^{2} ) \)
5$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
11$C_2^2$ \( 1 - 76 T - 155275 T^{2} - 76 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
19$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
23$C_2^2$ \( 1 - 4952 T + 18085961 T^{2} - 4952 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 7282 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 8886 T + 9617039 T^{2} - 8886 p^{5} T^{3} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 11748 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 + 24550 T + 184507007 T^{2} + 24550 p^{5} T^{3} + p^{10} T^{4} \)
59$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
67$C_2^2$ \( 1 + 69364 T + 3461239389 T^{2} + 69364 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 2224 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
79$C_2^2$ \( 1 + 80168 T + 3349851825 T^{2} + 80168 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
97$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
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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

−15.02803088170051434083531658119, −14.25487887236242164915534054361, −13.83402098173615237309563946479, −12.94075601727643739658634060184, −12.76032305916006270488256223925, −11.80890450617666156216612863444, −10.79188271225877721964154660048, −10.66600418153605608773764310597, −9.889417980539586984598260024936, −9.499418099516848113232982554769, −8.676409011842994005936267099199, −8.571915459378809512632589738251, −7.50592385933956759888849705418, −7.17909803497092025660594220749, −6.28676626806682348573626347119, −4.68982984388865013993011455444, −4.54420280589619847263599005074, −2.85586637449605103085036914909, −1.23887419323159435713920577928, −0.817019250532297601689552575162, 0.817019250532297601689552575162, 1.23887419323159435713920577928, 2.85586637449605103085036914909, 4.54420280589619847263599005074, 4.68982984388865013993011455444, 6.28676626806682348573626347119, 7.17909803497092025660594220749, 7.50592385933956759888849705418, 8.571915459378809512632589738251, 8.676409011842994005936267099199, 9.499418099516848113232982554769, 9.889417980539586984598260024936, 10.66600418153605608773764310597, 10.79188271225877721964154660048, 11.80890450617666156216612863444, 12.76032305916006270488256223925, 12.94075601727643739658634060184, 13.83402098173615237309563946479, 14.25487887236242164915534054361, 15.02803088170051434083531658119

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