Properties

Label 4-4864e2-1.1-c1e2-0-0
Degree $4$
Conductor $23658496$
Sign $1$
Analytic cond. $1508.48$
Root an. cond. $6.23211$
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  + 2·3-s − 4·5-s − 2·7-s − 3·9-s − 8·11-s + 2·13-s − 8·15-s − 2·17-s + 2·19-s − 4·21-s + 6·23-s + 2·25-s − 14·27-s + 2·29-s + 4·31-s − 16·33-s + 8·35-s − 4·37-s + 4·39-s − 4·43-s + 12·45-s + 49-s − 4·51-s + 18·53-s + 32·55-s + 4·57-s − 6·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.755·7-s − 9-s − 2.41·11-s + 0.554·13-s − 2.06·15-s − 0.485·17-s + 0.458·19-s − 0.872·21-s + 1.25·23-s + 2/5·25-s − 2.69·27-s + 0.371·29-s + 0.718·31-s − 2.78·33-s + 1.35·35-s − 0.657·37-s + 0.640·39-s − 0.609·43-s + 1.78·45-s + 1/7·49-s − 0.560·51-s + 2.47·53-s + 4.31·55-s + 0.529·57-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23658496\)    =    \(2^{16} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1508.48\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23658496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041282284\)
\(L(\frac12)\) \(\approx\) \(1.041282284\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.409844142482034624542865921103, −8.160352193197529167691406961719, −7.82997708304346210022311066950, −7.51076617999811505742478471969, −7.15680808115473336038404330626, −6.95558838694580589641188612609, −6.13451387595584215331300907883, −5.98409177660554147615614263121, −5.46671440922679715753527619258, −5.15334466556711035579773083577, −4.68593268942407830191418814817, −4.32028731814252122288459318747, −3.57680945655732494895191467556, −3.51893930683257661830686548440, −3.06474870964278006041252469469, −2.92787557650664609175165027571, −2.21907037782180396589358294394, −2.10326835699938223521861625043, −0.68732342905364201773801909327, −0.38113433795143368092123168877, 0.38113433795143368092123168877, 0.68732342905364201773801909327, 2.10326835699938223521861625043, 2.21907037782180396589358294394, 2.92787557650664609175165027571, 3.06474870964278006041252469469, 3.51893930683257661830686548440, 3.57680945655732494895191467556, 4.32028731814252122288459318747, 4.68593268942407830191418814817, 5.15334466556711035579773083577, 5.46671440922679715753527619258, 5.98409177660554147615614263121, 6.13451387595584215331300907883, 6.95558838694580589641188612609, 7.15680808115473336038404330626, 7.51076617999811505742478471969, 7.82997708304346210022311066950, 8.160352193197529167691406961719, 8.409844142482034624542865921103

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