Properties

Label 2-637-1.1-c1-0-25
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  − 1.21·2-s + 1.74·3-s − 0.534·4-s − 2.21·5-s − 2.11·6-s + 3.06·8-s + 0.0444·9-s + 2.67·10-s − 0.789·11-s − 0.931·12-s + 13-s − 3.85·15-s − 2.64·16-s + 1.74·17-s − 0.0537·18-s − 4.32·19-s + 1.18·20-s + 0.955·22-s − 1.11·23-s + 5.35·24-s − 0.112·25-s − 1.21·26-s − 5.15·27-s − 8.48·29-s + 4.67·30-s − 5.70·31-s − 2.93·32-s + ⋯
L(s)  = 1  − 0.856·2-s + 1.00·3-s − 0.267·4-s − 0.988·5-s − 0.862·6-s + 1.08·8-s + 0.0148·9-s + 0.846·10-s − 0.237·11-s − 0.269·12-s + 0.277·13-s − 0.995·15-s − 0.661·16-s + 0.423·17-s − 0.0126·18-s − 0.991·19-s + 0.264·20-s + 0.203·22-s − 0.231·23-s + 1.09·24-s − 0.0225·25-s − 0.237·26-s − 0.992·27-s − 1.57·29-s + 0.852·30-s − 1.02·31-s − 0.518·32-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 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 - 1.74T + 3T^{2} \)
5 \( 1 + 2.21T + 5T^{2} \)
11 \( 1 + 0.789T + 11T^{2} \)
17 \( 1 - 1.74T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 + 8.74T + 47T^{2} \)
53 \( 1 - 7.95T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 - 6.55T + 67T^{2} \)
71 \( 1 - 5.85T + 71T^{2} \)
73 \( 1 + 8.00T + 73T^{2} \)
79 \( 1 + 6.91T + 79T^{2} \)
83 \( 1 + 3.14T + 83T^{2} \)
89 \( 1 + 3.39T + 89T^{2} \)
97 \( 1 - 0.0981T + 97T^{2} \)
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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

−9.886448827782354447207839573212, −9.092376386554425210593750565535, −8.372852030371903580352969069313, −7.900006604580244668542591018722, −7.11216675889706320106399394130, −5.53414056094829906550762555246, −4.17936796705809210750168451694, −3.47344476654452636802043916797, −1.94168055883410583124853330460, 0, 1.94168055883410583124853330460, 3.47344476654452636802043916797, 4.17936796705809210750168451694, 5.53414056094829906550762555246, 7.11216675889706320106399394130, 7.900006604580244668542591018722, 8.372852030371903580352969069313, 9.092376386554425210593750565535, 9.886448827782354447207839573212

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