Properties

Label 4-2738e2-1.1-c1e2-0-4
Degree $4$
Conductor $7496644$
Sign $1$
Analytic cond. $477.992$
Root an. cond. $4.67579$
Motivic weight $1$
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  + 2·2-s + 3·3-s + 3·4-s + 5-s + 6·6-s + 2·7-s + 4·8-s + 4·9-s + 2·10-s − 11-s + 9·12-s + 13-s + 4·14-s + 3·15-s + 5·16-s + 12·17-s + 8·18-s − 4·19-s + 3·20-s + 6·21-s − 2·22-s + 3·23-s + 12·24-s − 6·25-s + 2·26-s + 6·27-s + 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s + 0.755·7-s + 1.41·8-s + 4/3·9-s + 0.632·10-s − 0.301·11-s + 2.59·12-s + 0.277·13-s + 1.06·14-s + 0.774·15-s + 5/4·16-s + 2.91·17-s + 1.88·18-s − 0.917·19-s + 0.670·20-s + 1.30·21-s − 0.426·22-s + 0.625·23-s + 2.44·24-s − 6/5·25-s + 0.392·26-s + 1.15·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7496644\)    =    \(2^{2} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(477.992\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2738} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(19.16770422\)
\(L(\frac12)\) \(\approx\) \(19.16770422\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
37 \( 1 \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 21 T + 253 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 7 T + 11 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.944968886000003926650433398828, −8.603421792293442815607614220204, −7.941712339654222058227939672443, −7.86056068938729605223363819641, −7.56462152729171433762411344891, −7.46850969697944667830951199919, −6.46326724060365295881527750344, −6.37499345664603451757224648053, −5.83414971611436273708895190435, −5.42676202893680327516757044499, −5.11805747079309522574553299921, −4.72031344853855943081692986728, −4.02019165312138306676204079704, −3.80025024914192379694990640252, −3.39755323657835636010866385661, −2.96548774308850045829590794528, −2.52714308561159724942380146071, −2.13669863292038347831544081484, −1.54305188307646409208210980611, −1.06920297575131178035674608636, 1.06920297575131178035674608636, 1.54305188307646409208210980611, 2.13669863292038347831544081484, 2.52714308561159724942380146071, 2.96548774308850045829590794528, 3.39755323657835636010866385661, 3.80025024914192379694990640252, 4.02019165312138306676204079704, 4.72031344853855943081692986728, 5.11805747079309522574553299921, 5.42676202893680327516757044499, 5.83414971611436273708895190435, 6.37499345664603451757224648053, 6.46326724060365295881527750344, 7.46850969697944667830951199919, 7.56462152729171433762411344891, 7.86056068938729605223363819641, 7.941712339654222058227939672443, 8.603421792293442815607614220204, 8.944968886000003926650433398828

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