Properties

Label 4-1078e2-1.1-c1e2-0-31
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
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 + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 4·8-s + 2·9-s − 4·10-s + 2·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s + 4·17-s + 4·18-s + 10·19-s − 6·20-s + 4·22-s + 8·23-s + 8·24-s − 2·25-s + 4·26-s + 6·27-s − 8·30-s − 4·31-s + 6·32-s + 4·33-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.41·8-s + 2/3·9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 0.942·18-s + 2.29·19-s − 1.34·20-s + 0.852·22-s + 1.66·23-s + 1.63·24-s − 2/5·25-s + 0.784·26-s + 1.15·27-s − 1.46·30-s − 0.718·31-s + 1.06·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.840023712\)
\(L(\frac12)\) \(\approx\) \(8.840023712\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−10.00829958563360915913838911374, −9.743913546027839964224405485591, −9.142548431920584778533938054409, −8.868440804206357708140504609090, −8.195523686917400799090646390594, −8.107011218752175823817849183982, −7.42804411366250484970207537346, −7.03886884663017761430529089934, −7.03000964499145678562181388468, −6.27285384438401664958786921833, −5.60433448207109467768543833834, −5.28717239674767381762556781599, −4.89768914171678265713289991347, −4.24663132650566531859521090082, −3.69541219660826411839179615129, −3.43146858905407759239798601585, −3.08791371561946288993837862507, −2.65054860399318456182246322213, −1.55719080927706853210503358377, −1.16174622122432376623073397048, 1.16174622122432376623073397048, 1.55719080927706853210503358377, 2.65054860399318456182246322213, 3.08791371561946288993837862507, 3.43146858905407759239798601585, 3.69541219660826411839179615129, 4.24663132650566531859521090082, 4.89768914171678265713289991347, 5.28717239674767381762556781599, 5.60433448207109467768543833834, 6.27285384438401664958786921833, 7.03000964499145678562181388468, 7.03886884663017761430529089934, 7.42804411366250484970207537346, 8.107011218752175823817849183982, 8.195523686917400799090646390594, 8.868440804206357708140504609090, 9.142548431920584778533938054409, 9.743913546027839964224405485591, 10.00829958563360915913838911374

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