Properties

Degree $4$
Conductor $39690000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 12·19-s + 2·29-s − 4·31-s + 16·41-s − 49-s + 20·59-s + 18·71-s − 2·79-s + 4·89-s + 24·101-s − 18·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 0.603·11-s − 2.75·19-s + 0.371·29-s − 0.718·31-s + 2.49·41-s − 1/7·49-s + 2.60·59-s + 2.13·71-s − 0.225·79-s + 0.423·89-s + 2.38·101-s − 1.72·109-s − 1.72·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.445176074\)
\(L(\frac12)\) \(\approx\) \(2.445176074\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.218669316908950529251621970975, −7.934720996971580467658431262459, −7.51244722779132455488311938850, −7.12350791899728955739604573591, −6.64725579999132615347491504221, −6.57740782418473537831566035843, −6.10270164826179713246749413678, −5.86906162130743002115587692141, −5.35323659869098991917157244386, −4.99241485170495571128446592611, −4.52020567753394969491021155998, −4.16033409312504071078889926326, −3.82288345560584941836361164037, −3.68640903147524051846045871005, −2.83445810712559253034214697924, −2.49197133748881539672909304268, −2.10422136582708275422063904289, −1.72636298507664953735269052166, −0.920772175312508306128775001377, −0.46438957403768265157818910288, 0.46438957403768265157818910288, 0.920772175312508306128775001377, 1.72636298507664953735269052166, 2.10422136582708275422063904289, 2.49197133748881539672909304268, 2.83445810712559253034214697924, 3.68640903147524051846045871005, 3.82288345560584941836361164037, 4.16033409312504071078889926326, 4.52020567753394969491021155998, 4.99241485170495571128446592611, 5.35323659869098991917157244386, 5.86906162130743002115587692141, 6.10270164826179713246749413678, 6.57740782418473537831566035843, 6.64725579999132615347491504221, 7.12350791899728955739604573591, 7.51244722779132455488311938850, 7.934720996971580467658431262459, 8.218669316908950529251621970975

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