Properties

Label 4-3450e2-1.1-c1e2-0-12
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 4·11-s + 16-s + 16·19-s − 8·29-s + 36-s − 4·41-s + 4·44-s + 14·49-s − 16·59-s + 4·61-s − 64-s + 20·71-s − 16·76-s + 16·79-s + 81-s + 24·89-s + 4·99-s + 24·101-s + 20·109-s + 8·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 3.67·19-s − 1.48·29-s + 1/6·36-s − 0.624·41-s + 0.603·44-s + 2·49-s − 2.08·59-s + 0.512·61-s − 1/8·64-s + 2.37·71-s − 1.83·76-s + 1.80·79-s + 1/9·81-s + 2.54·89-s + 0.402·99-s + 2.38·101-s + 1.91·109-s + 0.742·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.198510304\)
\(L(\frac12)\) \(\approx\) \(2.198510304\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−9.108792932344571634927519014254, −8.323516246670792470470583458685, −7.906007519820990042421048372561, −7.64364110372990049015430630430, −7.34789011723112045438592156963, −7.29623657180821848906325632604, −6.33198267606459058781549834071, −6.23059551650532439949833130639, −5.58876043468952255389389342209, −5.20999647729956396555254722645, −5.10195536559024912931784383956, −4.90337421142677275128985205570, −3.87611049359390698065887618396, −3.79431825527151714091635283865, −3.08154558190132171322615153754, −3.07629251146014194041135800566, −2.27947879961208929765099071462, −1.80728213568130872542643341006, −0.945099906505614831662834854500, −0.56485119263180306817068232408, 0.56485119263180306817068232408, 0.945099906505614831662834854500, 1.80728213568130872542643341006, 2.27947879961208929765099071462, 3.07629251146014194041135800566, 3.08154558190132171322615153754, 3.79431825527151714091635283865, 3.87611049359390698065887618396, 4.90337421142677275128985205570, 5.10195536559024912931784383956, 5.20999647729956396555254722645, 5.58876043468952255389389342209, 6.23059551650532439949833130639, 6.33198267606459058781549834071, 7.29623657180821848906325632604, 7.34789011723112045438592156963, 7.64364110372990049015430630430, 7.906007519820990042421048372561, 8.323516246670792470470583458685, 9.108792932344571634927519014254

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