Properties

Label 4-913952-1.1-c1e2-0-1
Degree $4$
Conductor $913952$
Sign $1$
Analytic cond. $58.2743$
Root an. cond. $2.76292$
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  + 2-s + 4-s + 6·5-s + 8-s − 5·9-s + 6·10-s + 16-s − 6·17-s − 5·18-s + 6·20-s + 17·25-s + 12·29-s + 32-s − 6·34-s − 5·36-s + 14·37-s + 6·40-s − 30·45-s − 13·49-s + 17·50-s + 12·58-s + 16·61-s + 64-s − 6·68-s − 5·72-s − 4·73-s + 14·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 2.68·5-s + 0.353·8-s − 5/3·9-s + 1.89·10-s + 1/4·16-s − 1.45·17-s − 1.17·18-s + 1.34·20-s + 17/5·25-s + 2.22·29-s + 0.176·32-s − 1.02·34-s − 5/6·36-s + 2.30·37-s + 0.948·40-s − 4.47·45-s − 1.85·49-s + 2.40·50-s + 1.57·58-s + 2.04·61-s + 1/8·64-s − 0.727·68-s − 0.589·72-s − 0.468·73-s + 1.62·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 913952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 913952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(913952\)    =    \(2^{5} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58.2743\)
Root analytic conductor: \(2.76292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 913952,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.967635186\)
\(L(\frac12)\) \(\approx\) \(4.967635186\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.244134846950840981107676678567, −7.85100333312502477176899037562, −6.92421875375835189696702665586, −6.43741307804194710670707219872, −6.40028526578024760134946435749, −5.93814316414134510481853694029, −5.56442463767257268744193982616, −5.11332595609013512036605695413, −4.68671862775252250000074194944, −4.14534392918182308978597121442, −3.04804615759780815838301814050, −2.86630826934752796258245605486, −2.19431989234797768682378556675, −2.01914482841960457865874156726, −0.925636388633133141442280821607, 0.925636388633133141442280821607, 2.01914482841960457865874156726, 2.19431989234797768682378556675, 2.86630826934752796258245605486, 3.04804615759780815838301814050, 4.14534392918182308978597121442, 4.68671862775252250000074194944, 5.11332595609013512036605695413, 5.56442463767257268744193982616, 5.93814316414134510481853694029, 6.40028526578024760134946435749, 6.43741307804194710670707219872, 6.92421875375835189696702665586, 7.85100333312502477176899037562, 8.244134846950840981107676678567

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