Properties

Label 2-8624-1.1-c1-0-172
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
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  + 2·3-s − 2·5-s + 9-s − 11-s + 2·13-s − 4·15-s + 2·19-s − 25-s − 4·27-s + 6·29-s − 4·31-s − 2·33-s + 2·37-s + 4·39-s + 8·41-s − 12·43-s − 2·45-s − 12·47-s − 2·53-s + 2·55-s + 4·57-s − 10·59-s − 10·61-s − 4·65-s + 12·67-s − 4·71-s + 12·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.03·15-s + 0.458·19-s − 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.640·39-s + 1.24·41-s − 1.82·43-s − 0.298·45-s − 1.75·47-s − 0.274·53-s + 0.269·55-s + 0.529·57-s − 1.30·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.474·71-s + 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8624\)    =    \(2^{4} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(68.8629\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8624,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82619487448678845685486744928, −6.88420472227818731407746511896, −6.17954231644219072510468276327, −5.21957456919730061256036186751, −4.46334868900367902524257023299, −3.60662757255630072193242606933, −3.23400828637191869712477583111, −2.37777618848739236526734684860, −1.37085742552661853239671743807, 0, 1.37085742552661853239671743807, 2.37777618848739236526734684860, 3.23400828637191869712477583111, 3.60662757255630072193242606933, 4.46334868900367902524257023299, 5.21957456919730061256036186751, 6.17954231644219072510468276327, 6.88420472227818731407746511896, 7.82619487448678845685486744928

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