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Properties

Label 2-637-91.74-c1-0-21
Degree $2$
Conductor $637$
Sign $0.326 - 0.945i$
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  + 1.85·2-s + (1.14 + 1.98i)3-s + 1.45·4-s + (−0.0986 − 0.170i)5-s + (2.13 + 3.69i)6-s − 1.01·8-s + (−1.13 + 1.95i)9-s + (−0.183 − 0.317i)10-s + (2.09 + 3.62i)11-s + (1.66 + 2.88i)12-s + (2.72 + 2.36i)13-s + (0.226 − 0.392i)15-s − 4.79·16-s − 0.841·17-s + (−2.10 + 3.64i)18-s + (0.675 − 1.17i)19-s + ⋯
L(s)  = 1  + 1.31·2-s + (0.662 + 1.14i)3-s + 0.726·4-s + (−0.0441 − 0.0764i)5-s + (0.870 + 1.50i)6-s − 0.359·8-s + (−0.377 + 0.653i)9-s + (−0.0579 − 0.100i)10-s + (0.630 + 1.09i)11-s + (0.481 + 0.833i)12-s + (0.755 + 0.655i)13-s + (0.0584 − 0.101i)15-s − 1.19·16-s − 0.204·17-s + (−0.495 + 0.858i)18-s + (0.155 − 0.268i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76310 + 1.96802i\)
\(L(\frac12)\) \(\approx\) \(2.76310 + 1.96802i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.72 - 2.36i)T \)
good2 \( 1 - 1.85T + 2T^{2} \)
3 \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.0986 + 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.841T + 17T^{2} \)
19 \( 1 + (-0.675 + 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.11T + 23T^{2} \)
29 \( 1 + (-4.11 + 7.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.640 - 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + (-2.69 + 4.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.66 + 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.83 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.05T + 59T^{2} \)
61 \( 1 + (5.68 - 9.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.69 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.98 - 5.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.94 + 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-9.73 - 16.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.76589926642652659522782672458, −9.806744252244703481406170170786, −9.157619770806648842696099015142, −8.347550413156564810009952289595, −6.87880661546915129576860941141, −6.05258996623747266684443541089, −4.71266604731963780440994132550, −4.28984917111468093101783385148, −3.52683143997779211880704356583, −2.28768504223823744189439885375, 1.35379884565889016145538646438, 2.92934278500421617752703630122, 3.50589051999421539977626693469, 4.83566968034486254794606180293, 6.07070325171585040219128455524, 6.48482440125680120288174421588, 7.70398477176143873328067776235, 8.473347051660821810571517236639, 9.285364602641729560173425186695, 10.80299919888429879540466033297

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