Properties

Label 4-3450e2-1.1-c1e2-0-28
Degree $4$
Conductor $11902500$
Sign $1$
Analytic cond. $758.913$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 6·11-s + 16-s + 6·19-s − 6·29-s − 20·31-s + 36-s − 18·41-s + 6·44-s + 5·49-s − 16·59-s + 24·61-s − 64-s − 28·71-s − 6·76-s + 34·79-s + 81-s + 12·89-s + 6·99-s − 12·101-s + 6·116-s + 5·121-s + 20·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 1.37·19-s − 1.11·29-s − 3.59·31-s + 1/6·36-s − 2.81·41-s + 0.904·44-s + 5/7·49-s − 2.08·59-s + 3.07·61-s − 1/8·64-s − 3.32·71-s − 0.688·76-s + 3.82·79-s + 1/9·81-s + 1.27·89-s + 0.603·99-s − 1.19·101-s + 0.557·116-s + 5/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11902500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(758.913\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.594727561096250568607914683950, −7.947742416903719463836030478211, −7.60604083458021867815907882228, −7.42642995809189757533603819840, −7.12240693561778877532409930352, −6.58193992554733461097521977922, −6.00909156403341296823250949620, −5.54860256187141924715838265477, −5.39423309486182384637904724093, −4.95355972115712410778498567211, −4.95156745147568085045581984651, −3.87948129659742253858868981963, −3.77193710054615342007457735688, −3.24398825784547830860069987967, −2.95670951281968131037554877040, −2.09687067171718710887999088900, −1.96010050643571310223846467719, −1.13531973161842769993227485333, 0, 0, 1.13531973161842769993227485333, 1.96010050643571310223846467719, 2.09687067171718710887999088900, 2.95670951281968131037554877040, 3.24398825784547830860069987967, 3.77193710054615342007457735688, 3.87948129659742253858868981963, 4.95156745147568085045581984651, 4.95355972115712410778498567211, 5.39423309486182384637904724093, 5.54860256187141924715838265477, 6.00909156403341296823250949620, 6.58193992554733461097521977922, 7.12240693561778877532409930352, 7.42642995809189757533603819840, 7.60604083458021867815907882228, 7.947742416903719463836030478211, 8.594727561096250568607914683950

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