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Properties

Label 2-637-7.4-c1-0-31
Degree $2$
Conductor $637$
Sign $0.900 + 0.435i$
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  + (0.588 + 1.01i)2-s + (1.67 − 2.90i)3-s + (0.308 − 0.533i)4-s + (1.57 + 2.72i)5-s + 3.94·6-s + 3.07·8-s + (−4.11 − 7.12i)9-s + (−1.85 + 3.20i)10-s + (0.386 − 0.669i)11-s + (−1.03 − 1.78i)12-s − 13-s + 10.5·15-s + (1.19 + 2.06i)16-s + (−2.87 + 4.98i)17-s + (4.83 − 8.38i)18-s + (−0.611 − 1.05i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.720i)2-s + (0.967 − 1.67i)3-s + (0.154 − 0.266i)4-s + (0.704 + 1.21i)5-s + 1.60·6-s + 1.08·8-s + (−1.37 − 2.37i)9-s + (−0.585 + 1.01i)10-s + (0.116 − 0.201i)11-s + (−0.298 − 0.516i)12-s − 0.277·13-s + 2.72·15-s + (0.298 + 0.516i)16-s + (−0.697 + 1.20i)17-s + (1.14 − 1.97i)18-s + (−0.140 − 0.242i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88270 - 0.661254i\)
\(L(\frac12)\) \(\approx\) \(2.88270 - 0.661254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.588 - 1.01i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.67 + 2.90i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.57 - 2.72i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.386 + 0.669i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.87 - 4.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.611 + 1.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.49 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + (3.06 - 5.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.49 + 4.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 + 2.73T + 43T^{2} \)
47 \( 1 + (-2.68 - 4.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.89 + 8.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.25 + 2.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.16 - 3.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (2.58 - 4.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.271 - 0.469i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-4.61 - 7.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.26T + 97T^{2} \)
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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

−10.61115729617675603293422268739, −9.465149963844054735312610902379, −8.458846313270456170985010642466, −7.50466951730622590299588007797, −6.85144515537506227261058178051, −6.39731184263339212083275989287, −5.57650120181352931698054555442, −3.62247187725756920003582783014, −2.41639395086857898846195258357, −1.63388226194346806761296036425, 2.05415474336061865201920730023, 2.94770265265342989769607153119, 4.16148743751562304029694319533, 4.67743245249855001822238429271, 5.49538150347177319738532223989, 7.38799817246054000915243844587, 8.468389069415162227453078279944, 9.109361170913968095149314092274, 9.744017161548606273303197717438, 10.52497014344781328831634305083

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