Properties

Label 4-1148e2-1.1-c1e2-0-6
Degree $4$
Conductor $1317904$
Sign $-1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 2·9-s + 2·11-s + 16-s − 2·18-s − 2·22-s − 4·23-s + 2·25-s + 2·29-s − 32-s + 2·36-s − 4·37-s + 8·43-s + 2·44-s + 4·46-s − 7·49-s − 2·50-s − 10·53-s − 2·58-s + 64-s − 6·67-s + 4·71-s − 2·72-s + 4·74-s − 5·81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 0.603·11-s + 1/4·16-s − 0.471·18-s − 0.426·22-s − 0.834·23-s + 2/5·25-s + 0.371·29-s − 0.176·32-s + 1/3·36-s − 0.657·37-s + 1.21·43-s + 0.301·44-s + 0.589·46-s − 49-s − 0.282·50-s − 1.37·53-s − 0.262·58-s + 1/8·64-s − 0.733·67-s + 0.474·71-s − 0.235·72-s + 0.464·74-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 + p T^{2} \)
41$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
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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

−7.75879282270896903377551228368, −7.39578678415885792104275519763, −6.97556319921434535593897623868, −6.40638575684129168429288779053, −6.27064936742835988224087556021, −5.68452106093720067685647453040, −5.00240665962561188704553339995, −4.69328291528169407241091839361, −3.93221214628784589867356016281, −3.73297582042354673548602632973, −2.90985414229579052886258136156, −2.42217906983837693240322077516, −1.59618459644696579161799921540, −1.20477784251621165121279381575, 0, 1.20477784251621165121279381575, 1.59618459644696579161799921540, 2.42217906983837693240322077516, 2.90985414229579052886258136156, 3.73297582042354673548602632973, 3.93221214628784589867356016281, 4.69328291528169407241091839361, 5.00240665962561188704553339995, 5.68452106093720067685647453040, 6.27064936742835988224087556021, 6.40638575684129168429288779053, 6.97556319921434535593897623868, 7.39578678415885792104275519763, 7.75879282270896903377551228368

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