Properties

Label 2-637-1.1-c1-0-5
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 $0$

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: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6212752590\)
\(L(\frac12)\) \(\approx\) \(0.6212752590\)
\(L(\frac{3}{2})\) 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

−10.53343006831673055775511684164, −9.610213531392557777959888858748, −9.182642743920545154488125772139, −8.000652481642021648717183770220, −7.11318411145537296109878140346, −5.89252159464686171875496880292, −5.38399324176441662561470680126, −4.24210765989283840943816093661, −2.35238640756927560759022767398, −0.821424707671999522381674432919, 0.821424707671999522381674432919, 2.35238640756927560759022767398, 4.24210765989283840943816093661, 5.38399324176441662561470680126, 5.89252159464686171875496880292, 7.11318411145537296109878140346, 8.000652481642021648717183770220, 9.182642743920545154488125772139, 9.610213531392557777959888858748, 10.53343006831673055775511684164

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