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Properties

Label 2-637-91.51-c1-0-20
Degree $2$
Conductor $637$
Sign $0.286 + 0.957i$
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  + (−2.36 − 1.36i)2-s + (0.577 + 1.00i)3-s + (2.73 + 4.74i)4-s + (1.62 + 0.938i)5-s − 3.16i·6-s − 9.50i·8-s + (0.831 − 1.44i)9-s + (−2.56 − 4.44i)10-s + (1.99 − 1.14i)11-s + (−3.16 + 5.48i)12-s + (0.574 − 3.55i)13-s + 2.17i·15-s + (−7.51 + 13.0i)16-s + (−3.03 − 5.25i)17-s + (−3.93 + 2.27i)18-s + (−4.46 − 2.57i)19-s + ⋯
L(s)  = 1  + (−1.67 − 0.966i)2-s + (0.333 + 0.577i)3-s + (1.36 + 2.37i)4-s + (0.727 + 0.419i)5-s − 1.29i·6-s − 3.36i·8-s + (0.277 − 0.480i)9-s + (−0.811 − 1.40i)10-s + (0.600 − 0.346i)11-s + (−0.913 + 1.58i)12-s + (0.159 − 0.987i)13-s + 0.560i·15-s + (−1.87 + 3.25i)16-s + (−0.736 − 1.27i)17-s + (−0.928 + 0.536i)18-s + (−1.02 − 0.590i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.286 + 0.957i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.286 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680017 - 0.506179i\)
\(L(\frac12)\) \(\approx\) \(0.680017 - 0.506179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.574 + 3.55i)T \)
good2 \( 1 + (2.36 + 1.36i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.577 - 1.00i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.62 - 0.938i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.99 + 1.14i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.03 + 5.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.46 + 2.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.20 + 3.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 + (3.75 - 2.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.74 - 1.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 + (-7.71 - 4.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.05 + 7.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.18 - 3.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.00 - 6.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.00 + 0.578i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 + (1.70 - 0.984i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.90 - 3.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.49iT - 83T^{2} \)
89 \( 1 + (-9.03 - 5.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.51iT - 97T^{2} \)
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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

−10.44364562661767502286877212905, −9.492611518635773601792292278199, −8.992505184065548260413383822738, −8.296447783187427831944021982691, −7.01516307900526986399141451623, −6.39349029345563588826918001773, −4.40195870249600272824560262746, −3.12978433450251516388072071272, −2.44474136109754910769223916697, −0.808186139275833521948768010358, 1.54887218685790892134926459339, 1.97331270769842371319066403778, 4.56640407024469787329092175617, 5.93669746281344712685043734808, 6.53267990251252425176327474267, 7.35320037294913448903985917454, 8.250867821442921071812363564935, 8.906504660920988348971438400252, 9.537598079342370513766670965406, 10.43983515804464283239886117343

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