Properties

Label 2-8624-1.1-c1-0-129
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  − 3-s − 2·9-s + 11-s − 13-s − 6·17-s − 2·19-s + 6·23-s − 5·25-s + 5·27-s + 9·29-s + 4·31-s − 33-s + 2·37-s + 39-s − 6·41-s + 4·43-s + 6·47-s + 6·51-s + 2·57-s + 3·59-s + 11·61-s − 11·67-s − 6·69-s + 2·73-s + 5·75-s − 5·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 25-s + 0.962·27-s + 1.67·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.875·47-s + 0.840·51-s + 0.264·57-s + 0.390·59-s + 1.40·61-s − 1.34·67-s − 0.722·69-s + 0.234·73-s + 0.577·75-s − 0.562·79-s + 1/9·81-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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−7.23125371039698292443632360161, −6.65241472062653308222612416481, −6.14853718702059937895066567222, −5.34931737834486556960138497960, −4.66196038971614403585102351941, −4.05923614656913661394578603646, −2.91027296261784789907385109519, −2.33677433521244575835493915149, −1.07055595102177968762291052173, 0, 1.07055595102177968762291052173, 2.33677433521244575835493915149, 2.91027296261784789907385109519, 4.05923614656913661394578603646, 4.66196038971614403585102351941, 5.34931737834486556960138497960, 6.14853718702059937895066567222, 6.65241472062653308222612416481, 7.23125371039698292443632360161

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