Properties

Label 4-2736e2-1.1-c1e2-0-20
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  − 6·5-s − 6·17-s − 8·19-s + 17·25-s + 8·31-s + 11·49-s + 24·59-s + 14·61-s − 16·67-s + 24·71-s − 10·73-s + 16·79-s + 36·85-s + 48·95-s − 12·101-s + 16·103-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2.68·5-s − 1.45·17-s − 1.83·19-s + 17/5·25-s + 1.43·31-s + 11/7·49-s + 3.12·59-s + 1.79·61-s − 1.95·67-s + 2.84·71-s − 1.17·73-s + 1.80·79-s + 3.90·85-s + 4.92·95-s − 1.19·101-s + 1.57·103-s − 0.454·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 3.85·155-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9736903023\)
\(L(\frac12)\) \(\approx\) \(0.9736903023\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−8.677189307334523947668671563372, −8.363337226451050603817472399491, −8.277023210437421579829632390968, −8.234769601686634692742464933378, −7.32751736569034931640156212367, −7.27270882298199483092958874418, −6.71714998488573631124350837565, −6.69564614107282682641650223683, −5.95012603191619208721959744636, −5.58857720982145297832256257370, −4.70711830707886589617803915784, −4.68494692367126008833653161485, −4.14345795301477356084412913527, −3.97240210870153619274509494671, −3.56966940227906811305434824874, −3.01995029873567641473879516278, −2.21823605208598876950204611481, −2.16865445928831136851258342401, −0.73301545130877721427260601641, −0.49334480651321311363649195246, 0.49334480651321311363649195246, 0.73301545130877721427260601641, 2.16865445928831136851258342401, 2.21823605208598876950204611481, 3.01995029873567641473879516278, 3.56966940227906811305434824874, 3.97240210870153619274509494671, 4.14345795301477356084412913527, 4.68494692367126008833653161485, 4.70711830707886589617803915784, 5.58857720982145297832256257370, 5.95012603191619208721959744636, 6.69564614107282682641650223683, 6.71714998488573631124350837565, 7.27270882298199483092958874418, 7.32751736569034931640156212367, 8.234769601686634692742464933378, 8.277023210437421579829632390968, 8.363337226451050603817472399491, 8.677189307334523947668671563372

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