Properties

Label 4-6300e2-1.1-c1e2-0-10
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  − 6·11-s − 4·19-s − 18·29-s + 16·31-s − 49-s + 24·59-s + 16·61-s − 10·79-s + 24·89-s + 12·101-s + 38·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.80·11-s − 0.917·19-s − 3.34·29-s + 2.87·31-s − 1/7·49-s + 3.12·59-s + 2.04·61-s − 1.12·79-s + 2.54·89-s + 1.19·101-s + 3.63·109-s + 5/11·121-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-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

\(L(1)\) \(\approx\) \(2.168575963\)
\(L(\frac12)\) \(\approx\) \(2.168575963\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−8.097536460848612501766689778866, −7.995036569844709837871582790280, −7.36861687619956048761521613228, −7.36615797447158924154017608690, −6.83022625351355182490901155850, −6.49162882181503753863755006294, −5.88427825224716048352033220052, −5.86462622250061922150790372894, −5.24942638568418603780842995720, −5.16847347205826051430424425578, −4.54744527736157926293075129106, −4.35399722174269092243911263671, −3.61465765061589448717294852684, −3.61411530680295862166639123517, −2.87705427227391922406622209807, −2.52185719193018361747496175556, −1.98701765255630521234778684789, −1.94285307474004409515243035418, −0.75084371025582199638616592370, −0.50458200753257436089515812021, 0.50458200753257436089515812021, 0.75084371025582199638616592370, 1.94285307474004409515243035418, 1.98701765255630521234778684789, 2.52185719193018361747496175556, 2.87705427227391922406622209807, 3.61411530680295862166639123517, 3.61465765061589448717294852684, 4.35399722174269092243911263671, 4.54744527736157926293075129106, 5.16847347205826051430424425578, 5.24942638568418603780842995720, 5.86462622250061922150790372894, 5.88427825224716048352033220052, 6.49162882181503753863755006294, 6.83022625351355182490901155850, 7.36615797447158924154017608690, 7.36861687619956048761521613228, 7.995036569844709837871582790280, 8.097536460848612501766689778866

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