Properties

Label 4-7942e2-1.1-c1e2-0-1
Degree $4$
Conductor $63075364$
Sign $1$
Analytic cond. $4021.73$
Root an. cond. $7.96349$
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  + 2·2-s − 3-s + 3·4-s + 4·5-s − 2·6-s + 3·7-s + 4·8-s − 9-s + 8·10-s + 2·11-s − 3·12-s + 3·13-s + 6·14-s − 4·15-s + 5·16-s + 3·17-s − 2·18-s + 12·20-s − 3·21-s + 4·22-s + 7·23-s − 4·24-s + 2·25-s + 6·26-s + 9·28-s − 29-s − 8·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s + 1.78·5-s − 0.816·6-s + 1.13·7-s + 1.41·8-s − 1/3·9-s + 2.52·10-s + 0.603·11-s − 0.866·12-s + 0.832·13-s + 1.60·14-s − 1.03·15-s + 5/4·16-s + 0.727·17-s − 0.471·18-s + 2.68·20-s − 0.654·21-s + 0.852·22-s + 1.45·23-s − 0.816·24-s + 2/5·25-s + 1.17·26-s + 1.70·28-s − 0.185·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63075364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63075364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63075364\)    =    \(2^{2} \cdot 11^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(4021.73\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7942} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63075364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.49046773\)
\(L(\frac12)\) \(\approx\) \(15.49046773\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
19 \( 1 \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T + 98 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−7.76149139709409670564370199030, −7.49656327108103711519974684412, −7.30739420095091067637585896249, −6.77258882591034770576810068363, −6.26152210085320616205618021448, −6.20922550054195113709633633278, −5.87130281168689492098727740929, −5.58312156423369932347845145021, −5.20909408360791196359647595809, −5.05917114168770782977842868853, −4.64430177490141412929841993326, −4.16835199699085956052137885388, −3.68183590469577613779774575966, −3.51966767263986573898780279740, −2.72051812345261325163503029859, −2.66834210528100327403904088613, −1.84362520720394966377157797982, −1.73505123945761296776573171653, −1.32574849154556456165089571751, −0.72496175532024712643623898876, 0.72496175532024712643623898876, 1.32574849154556456165089571751, 1.73505123945761296776573171653, 1.84362520720394966377157797982, 2.66834210528100327403904088613, 2.72051812345261325163503029859, 3.51966767263986573898780279740, 3.68183590469577613779774575966, 4.16835199699085956052137885388, 4.64430177490141412929841993326, 5.05917114168770782977842868853, 5.20909408360791196359647595809, 5.58312156423369932347845145021, 5.87130281168689492098727740929, 6.20922550054195113709633633278, 6.26152210085320616205618021448, 6.77258882591034770576810068363, 7.30739420095091067637585896249, 7.49656327108103711519974684412, 7.76149139709409670564370199030

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