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Properties

Label 2-637-91.88-c1-0-27
Degree $2$
Conductor $637$
Sign $-0.0780 + 0.996i$
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  + (−1.82 − 1.05i)2-s + 2.26·3-s + (1.22 + 2.12i)4-s + (3.11 − 1.80i)5-s + (−4.13 − 2.38i)6-s − 0.948i·8-s + 2.11·9-s − 7.59·10-s + 0.886i·11-s + (2.76 + 4.79i)12-s + (−1.17 − 3.40i)13-s + (7.05 − 4.07i)15-s + (1.44 − 2.51i)16-s + (−2.48 − 4.29i)17-s + (−3.86 − 2.23i)18-s − 2.37i·19-s + ⋯
L(s)  = 1  + (−1.29 − 0.745i)2-s + 1.30·3-s + (0.612 + 1.06i)4-s + (1.39 − 0.805i)5-s + (−1.68 − 0.973i)6-s − 0.335i·8-s + 0.705·9-s − 2.40·10-s + 0.267i·11-s + (0.799 + 1.38i)12-s + (−0.325 − 0.945i)13-s + (1.82 − 1.05i)15-s + (0.362 − 0.627i)16-s + (−0.601 − 1.04i)17-s + (−0.910 − 0.525i)18-s − 0.545i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0780 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0780 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0780 + 0.996i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0780 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977753 - 1.05733i\)
\(L(\frac12)\) \(\approx\) \(0.977753 - 1.05733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 3.40i)T \)
good2 \( 1 + (1.82 + 1.05i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 + (-3.11 + 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.886iT - 11T^{2} \)
17 \( 1 + (2.48 + 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.37iT - 19T^{2} \)
23 \( 1 + (1.92 - 3.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.640 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.33 - 4.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.34 - 4.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.4 + 6.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.58 + 1.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.46 - 4.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.34 - 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 - 8.42iT - 67T^{2} \)
71 \( 1 + (5.58 + 3.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.19 - 3.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.656i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.76iT - 83T^{2} \)
89 \( 1 + (3.13 + 1.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.401 + 0.231i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.808259522261073517327395033114, −9.555928618458034943635481131117, −8.885465652520451122481089984277, −8.174041591189213329222675465851, −7.33977347734365448215943538422, −5.81364831085623798002427971898, −4.69650659009447475738758573300, −2.84271668321121524128786644910, −2.34907576689520637293542276034, −1.10101692962524911214673654566, 1.78919787609118524732908592654, 2.62800958468126420482350938363, 4.09505698709778122844504712381, 6.08802632168083110763900214073, 6.41086589796768049760434119507, 7.57182780631567599617124030871, 8.257972591683322309991873122361, 9.146879526831753368121754666602, 9.578601154109642269129258455122, 10.28946254968236966459616531806

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