Properties

Label 2-8624-1.1-c1-0-190
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 1.41·5-s − 0.999·9-s + 11-s + 2.82·13-s + 2.00·15-s − 5.65·17-s − 2.82·19-s + 2·23-s − 2.99·25-s − 5.65·27-s − 6·29-s + 1.41·31-s + 1.41·33-s − 2·37-s + 4.00·39-s + 5.65·41-s − 4·43-s − 1.41·45-s − 4.24·47-s − 8.00·51-s − 12·53-s + 1.41·55-s − 4.00·57-s − 4.24·59-s − 5.65·61-s + 4.00·65-s + ⋯
L(s)  = 1  + 0.816·3-s + 0.632·5-s − 0.333·9-s + 0.301·11-s + 0.784·13-s + 0.516·15-s − 1.37·17-s − 0.648·19-s + 0.417·23-s − 0.599·25-s − 1.08·27-s − 1.11·29-s + 0.254·31-s + 0.246·33-s − 0.328·37-s + 0.640·39-s + 0.883·41-s − 0.609·43-s − 0.210·45-s − 0.618·47-s − 1.12·51-s − 1.64·53-s + 0.190·55-s − 0.529·57-s − 0.552·59-s − 0.724·61-s + 0.496·65-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: no
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 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 4.24T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−7.57876736028286376065700324215, −6.58457650196766177452272450964, −6.19857826851861701516336484992, −5.43056530756312923324311392347, −4.50153256994148336194726783437, −3.78276805804753880596918911825, −3.02267128261271388800815658486, −2.17502743322467007387981342736, −1.58126441160005025325976768153, 0, 1.58126441160005025325976768153, 2.17502743322467007387981342736, 3.02267128261271388800815658486, 3.78276805804753880596918911825, 4.50153256994148336194726783437, 5.43056530756312923324311392347, 6.19857826851861701516336484992, 6.58457650196766177452272450964, 7.57876736028286376065700324215

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