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Properties

Label 2-8624-1.1-c1-0-196
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 − 4·13-s + 4·15-s − 4·17-s + 4·23-s − 25-s − 4·27-s − 6·29-s + 10·31-s − 2·33-s − 6·37-s − 8·39-s − 4·41-s − 12·43-s + 2·45-s − 10·47-s − 8·51-s − 6·53-s − 2·55-s + 2·59-s − 8·65-s − 8·67-s + 8·69-s + 12·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.03·15-s − 0.970·17-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s + 1.79·31-s − 0.348·33-s − 0.986·37-s − 1.28·39-s − 0.624·41-s − 1.82·43-s + 0.298·45-s − 1.45·47-s − 1.12·51-s − 0.824·53-s − 0.269·55-s + 0.260·59-s − 0.992·65-s − 0.977·67-s + 0.963·69-s + 1.42·71-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.51633165496452066565735742912, −6.78596846568392118954662100559, −6.18749203490606167640127654989, −5.13699302665429628170294703719, −4.79415566372924421890047210418, −3.62980781323120854331945127407, −2.94439912019830745165849006977, −2.24215446749499935575895860873, −1.67915649184277622840650438512, 0, 1.67915649184277622840650438512, 2.24215446749499935575895860873, 2.94439912019830745165849006977, 3.62980781323120854331945127407, 4.79415566372924421890047210418, 5.13699302665429628170294703719, 6.18749203490606167640127654989, 6.78596846568392118954662100559, 7.51633165496452066565735742912

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