Properties

Label 2-3626-1.1-c1-0-111
Degree $2$
Conductor $3626$
Sign $-1$
Analytic cond. $28.9537$
Root an. cond. $5.38086$
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  + 2-s + 4-s + 8-s − 3·9-s + 16-s − 2.82·17-s − 3·18-s + 4.24·19-s − 6·23-s − 5·25-s − 6·29-s − 4.24·31-s + 32-s − 2.82·34-s − 3·36-s + 37-s + 4.24·38-s + 7.07·41-s − 4·43-s − 6·46-s − 2.82·47-s − 5·50-s − 6·58-s − 1.41·59-s − 2.82·61-s − 4.24·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.353·8-s − 9-s + 0.250·16-s − 0.685·17-s − 0.707·18-s + 0.973·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.762·31-s + 0.176·32-s − 0.485·34-s − 0.5·36-s + 0.164·37-s + 0.688·38-s + 1.10·41-s − 0.609·43-s − 0.884·46-s − 0.412·47-s − 0.707·50-s − 0.787·58-s − 0.184·59-s − 0.362·61-s − 0.538·62-s + 0.125·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3626\)    =    \(2 \cdot 7^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(28.9537\)
Root analytic conductor: \(5.38086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3626,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
37 \( 1 - T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.952932978275074682316466454396, −7.51467785242406479008944379454, −6.47407787691715178504049915516, −5.80867489247740207901158607553, −5.28990484688904571809568106469, −4.24413829548230577512165382412, −3.54452487667790804262588064127, −2.62938535497031803619662259325, −1.72911612108547849255571744329, 0, 1.72911612108547849255571744329, 2.62938535497031803619662259325, 3.54452487667790804262588064127, 4.24413829548230577512165382412, 5.28990484688904571809568106469, 5.80867489247740207901158607553, 6.47407787691715178504049915516, 7.51467785242406479008944379454, 7.952932978275074682316466454396

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