Properties

Label 4-392e2-1.1-c1e2-0-13
Degree $4$
Conductor $153664$
Sign $1$
Analytic cond. $9.79774$
Root an. cond. $1.76921$
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  − 2·5-s + 3·9-s + 4·11-s + 4·13-s + 6·17-s − 8·19-s + 5·25-s + 12·29-s − 8·31-s + 2·37-s + 4·41-s − 8·43-s − 6·45-s + 8·47-s − 6·53-s − 8·55-s + 6·61-s − 8·65-s + 4·67-s − 16·71-s − 10·73-s − 16·79-s + 16·83-s − 12·85-s + 6·89-s + 16·95-s − 12·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 9-s + 1.20·11-s + 1.10·13-s + 1.45·17-s − 1.83·19-s + 25-s + 2.22·29-s − 1.43·31-s + 0.328·37-s + 0.624·41-s − 1.21·43-s − 0.894·45-s + 1.16·47-s − 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.992·65-s + 0.488·67-s − 1.89·71-s − 1.17·73-s − 1.80·79-s + 1.75·83-s − 1.30·85-s + 0.635·89-s + 1.64·95-s − 1.21·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(153664\)    =    \(2^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(9.79774\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{392} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 153664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.748193723\)
\(L(\frac12)\) \(\approx\) \(1.748193723\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−11.48027091577279013636710047891, −11.19971378739028603905201432941, −10.40469223640186487596893712487, −10.39261740599788675139683158107, −9.852428124813606514694109476450, −9.069022381431254229018538250769, −8.629841428112019665094923856396, −8.567313669073813564430591680918, −7.64423184874793286340285178526, −7.50902632612095066162361842807, −6.69145821398429487272474581386, −6.46224666431223096370456714252, −5.94211590686605898519012397209, −5.12967271343095858206091713244, −4.36449030072195860083032903856, −4.14973688010729382898161935693, −3.57096991569393319230331323260, −2.91873216452083230725512049416, −1.70575969712221466509363245223, −1.00222359229110157409557222164, 1.00222359229110157409557222164, 1.70575969712221466509363245223, 2.91873216452083230725512049416, 3.57096991569393319230331323260, 4.14973688010729382898161935693, 4.36449030072195860083032903856, 5.12967271343095858206091713244, 5.94211590686605898519012397209, 6.46224666431223096370456714252, 6.69145821398429487272474581386, 7.50902632612095066162361842807, 7.64423184874793286340285178526, 8.567313669073813564430591680918, 8.629841428112019665094923856396, 9.069022381431254229018538250769, 9.852428124813606514694109476450, 10.39261740599788675139683158107, 10.40469223640186487596893712487, 11.19971378739028603905201432941, 11.48027091577279013636710047891

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