Properties

Label 2-637-1.1-c1-0-8
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  − 2.65·2-s − 2.39·3-s + 5.05·4-s − 3.65·5-s + 6.36·6-s − 8.10·8-s + 2.74·9-s + 9.70·10-s + 0.655·11-s − 12.1·12-s + 13-s + 8.75·15-s + 11.4·16-s − 2.39·17-s − 7.27·18-s + 2.70·19-s − 18.4·20-s − 1.74·22-s + 7.36·23-s + 19.4·24-s + 8.36·25-s − 2.65·26-s + 0.621·27-s − 0.208·29-s − 23.2·30-s + 1.13·31-s − 14.1·32-s + ⋯
L(s)  = 1  − 1.87·2-s − 1.38·3-s + 2.52·4-s − 1.63·5-s + 2.59·6-s − 2.86·8-s + 0.913·9-s + 3.06·10-s + 0.197·11-s − 3.49·12-s + 0.277·13-s + 2.26·15-s + 2.85·16-s − 0.581·17-s − 1.71·18-s + 0.620·19-s − 4.12·20-s − 0.371·22-s + 1.53·23-s + 3.96·24-s + 1.67·25-s − 0.520·26-s + 0.119·27-s − 0.0386·29-s − 4.24·30-s + 0.204·31-s − 2.49·32-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 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 - 0.655T + 11T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + 0.208T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 + 7.44T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 - 5.25T + 53T^{2} \)
59 \( 1 + 8.25T + 59T^{2} \)
61 \( 1 - 1.89T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 6.75T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 1.51T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−10.42302853466581500292080089867, −9.138262503150044837639437170910, −8.499226119756723269213361127937, −7.42914455428864615921675834751, −7.01531503002281907134948596800, −6.01080538658186059675657642303, −4.66124219223746332786962196405, −3.14489727817398470301905971745, −1.10596898171468342754065453783, 0, 1.10596898171468342754065453783, 3.14489727817398470301905971745, 4.66124219223746332786962196405, 6.01080538658186059675657642303, 7.01531503002281907134948596800, 7.42914455428864615921675834751, 8.499226119756723269213361127937, 9.138262503150044837639437170910, 10.42302853466581500292080089867

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