Properties

Label 4-6300e2-1.1-c1e2-0-15
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
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  + 10·11-s + 12·19-s − 6·29-s + 4·31-s + 8·41-s − 49-s − 28·59-s + 8·61-s + 26·71-s − 2·79-s + 20·89-s − 24·101-s − 18·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 3.01·11-s + 2.75·19-s − 1.11·29-s + 0.718·31-s + 1.24·41-s − 1/7·49-s − 3.64·59-s + 1.02·61-s + 3.08·71-s − 0.225·79-s + 2.11·89-s − 2.38·101-s − 1.72·109-s + 4.81·121-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(5.028304357\)
\(L(\frac12)\) \(\approx\) \(5.028304357\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.138592665505716980512275892740, −7.80610981275829910691130007785, −7.37299374591034687257841257239, −7.33186489534492067639835635274, −6.68262199789362118176107008857, −6.44725819563221507430860038235, −6.24162775929171592738667734361, −5.81904514044146906363998668313, −5.22325354546920482854497137262, −5.14795289949211478914435016503, −4.57010695260788212405901715127, −4.06785444015059974613175157658, −3.76456397122422043364875489455, −3.61368564429587258230263218559, −2.99785683174150506126616708228, −2.70507015729051199756097413181, −1.86902151719149211874258175452, −1.47705117685146803237647981580, −1.11468970196047933546284484118, −0.65527481898075890960967101756, 0.65527481898075890960967101756, 1.11468970196047933546284484118, 1.47705117685146803237647981580, 1.86902151719149211874258175452, 2.70507015729051199756097413181, 2.99785683174150506126616708228, 3.61368564429587258230263218559, 3.76456397122422043364875489455, 4.06785444015059974613175157658, 4.57010695260788212405901715127, 5.14795289949211478914435016503, 5.22325354546920482854497137262, 5.81904514044146906363998668313, 6.24162775929171592738667734361, 6.44725819563221507430860038235, 6.68262199789362118176107008857, 7.33186489534492067639835635274, 7.37299374591034687257841257239, 7.80610981275829910691130007785, 8.138592665505716980512275892740

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