Properties

Label 2-637-1.1-c1-0-40
Degree $2$
Conductor $637$
Sign $-1$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + 1.41·2-s + 1.41·3-s − 4.41·5-s + 2.00·6-s − 2.82·8-s − 0.999·9-s − 6.24·10-s − 4.24·11-s + 13-s − 6.24·15-s − 4.00·16-s + 1.41·17-s − 1.41·18-s − 1.24·19-s − 6·22-s − 0.171·23-s − 4·24-s + 14.4·25-s + 1.41·26-s − 5.65·27-s + 5.82·29-s − 8.82·30-s + 5.24·31-s − 6·33-s + 2.00·34-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.816·3-s − 1.97·5-s + 0.816·6-s − 0.999·8-s − 0.333·9-s − 1.97·10-s − 1.27·11-s + 0.277·13-s − 1.61·15-s − 1.00·16-s + 0.342·17-s − 0.333·18-s − 0.285·19-s − 1.27·22-s − 0.0357·23-s − 0.816·24-s + 2.89·25-s + 0.277·26-s − 1.08·27-s + 1.08·29-s − 1.61·30-s + 0.941·31-s − 1.04·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 - 1.41T + 2T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + 4.41T + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 + 0.171T + 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 - 5.24T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + 3.17T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 + 5.82T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 - 0.757T + 73T^{2} \)
79 \( 1 + 1.48T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 + 4.41T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−10.30645035020357371713520906536, −8.908967220311914474610397015752, −8.247083166332149019636656831159, −7.77990722361272211819462870738, −6.53993320905502797566897863101, −5.15904078703470480396436077311, −4.41651203737942914312023578712, −3.37369432541033221291342542597, −2.91685377672197224188381781902, 0, 2.91685377672197224188381781902, 3.37369432541033221291342542597, 4.41651203737942914312023578712, 5.15904078703470480396436077311, 6.53993320905502797566897863101, 7.77990722361272211819462870738, 8.247083166332149019636656831159, 8.908967220311914474610397015752, 10.30645035020357371713520906536

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