Properties

Label 2-637-49.8-c1-0-42
Degree $2$
Conductor $637$
Sign $0.332 + 0.943i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.324i)2-s + (−2.07 + 0.998i)3-s + (0.406 − 1.78i)4-s + (2.95 − 1.42i)5-s + (−0.860 − 0.414i)6-s + (2.39 − 1.13i)7-s + (1.43 − 0.689i)8-s + (1.43 − 1.79i)9-s + (1.22 + 0.590i)10-s + (−3.15 − 3.95i)11-s + (0.935 + 4.09i)12-s + (−0.623 − 0.781i)13-s + (0.986 + 0.483i)14-s + (−4.70 + 5.89i)15-s + (−2.69 − 1.30i)16-s + (0.170 + 0.749i)17-s + ⋯
L(s)  = 1  + (0.182 + 0.229i)2-s + (−1.19 + 0.576i)3-s + (0.203 − 0.890i)4-s + (1.32 − 0.636i)5-s + (−0.351 − 0.169i)6-s + (0.903 − 0.427i)7-s + (0.506 − 0.243i)8-s + (0.476 − 0.597i)9-s + (0.387 + 0.186i)10-s + (−0.950 − 1.19i)11-s + (0.270 + 1.18i)12-s + (−0.172 − 0.216i)13-s + (0.263 + 0.129i)14-s + (−1.21 + 1.52i)15-s + (−0.674 − 0.325i)16-s + (0.0414 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.332 + 0.943i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11000 - 0.785616i\)
\(L(\frac12)\) \(\approx\) \(1.11000 - 0.785616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.39 + 1.13i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-0.258 - 0.324i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (2.07 - 0.998i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (-2.95 + 1.42i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.15 + 3.95i)T + (-2.44 + 10.7i)T^{2} \)
17 \( 1 + (-0.170 - 0.749i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + (1.18 - 5.19i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.36 + 5.96i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 6.89T + 31T^{2} \)
37 \( 1 + (-0.332 - 1.45i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-2.62 + 1.26i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (0.824 + 0.397i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-7.89 - 9.90i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.30 + 5.71i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (4.68 + 2.25i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-2.78 - 12.1i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 4.44T + 67T^{2} \)
71 \( 1 + (0.427 - 1.87i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.56 + 6.98i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 4.91T + 79T^{2} \)
83 \( 1 + (-6.11 + 7.67i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-8.82 + 11.0i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−10.42339570708481658993534189538, −9.970578297506163347732786536509, −8.813748966817502417842675011905, −7.72444319643421553199722220724, −6.04186375960175134617702621995, −5.97791068529818905882711898949, −5.08252646227663916422971761556, −4.44275907710748163856789171541, −2.16240612918128184971958377145, −0.795653734736063549366492461728, 1.96024291894315455876234568330, 2.51151077142178764436661513729, 4.52623160188865338216542570259, 5.29267924224378197934964272722, 6.35746075700842715061861425931, 6.96886878054306292800590985463, 7.926948215939764230203120396749, 9.003427499735970014270470140365, 10.42563258735035530125639772327, 10.68101433650603647015681226126

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