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Properties

Label 2-637-91.33-c1-0-28
Degree $2$
Conductor $637$
Sign $0.560 + 0.828i$
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  + (0.0302 − 0.112i)2-s − 2.59i·3-s + (1.72 + 0.993i)4-s + (0.456 + 1.70i)5-s + (−0.293 − 0.0785i)6-s + (0.329 − 0.329i)8-s − 3.75·9-s + 0.206·10-s + (−1.38 + 1.38i)11-s + (2.58 − 4.47i)12-s + (1.85 − 3.09i)13-s + (4.43 − 1.18i)15-s + (1.95 + 3.39i)16-s + (2.13 − 3.70i)17-s + (−0.113 + 0.423i)18-s + (3.01 − 3.01i)19-s + ⋯
L(s)  = 1  + (0.0213 − 0.0797i)2-s − 1.50i·3-s + (0.860 + 0.496i)4-s + (0.204 + 0.762i)5-s + (−0.119 − 0.0320i)6-s + (0.116 − 0.116i)8-s − 1.25·9-s + 0.0651·10-s + (−0.417 + 0.417i)11-s + (0.745 − 1.29i)12-s + (0.515 − 0.857i)13-s + (1.14 − 0.306i)15-s + (0.489 + 0.848i)16-s + (0.518 − 0.898i)17-s + (−0.0267 + 0.0997i)18-s + (0.692 − 0.692i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.560 + 0.828i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71401 - 0.909448i\)
\(L(\frac12)\) \(\approx\) \(1.71401 - 0.909448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 + 3.09i)T \)
good2 \( 1 + (-0.0302 + 0.112i)T + (-1.73 - i)T^{2} \)
3 \( 1 + 2.59iT - 3T^{2} \)
5 \( 1 + (-0.456 - 1.70i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.38 - 1.38i)T - 11iT^{2} \)
17 \( 1 + (-2.13 + 3.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.01 + 3.01i)T - 19iT^{2} \)
23 \( 1 + (-5.53 + 3.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.57 - 6.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.13 + 1.10i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.73 - 0.732i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.94 - 11.0i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.55 - 0.896i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.40 - 1.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.13 + 3.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.62 - 0.436i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 3.08iT - 61T^{2} \)
67 \( 1 + (-0.0139 - 0.0139i)T + 67iT^{2} \)
71 \( 1 + (-1.23 + 4.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.255 - 0.954i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.96 + 5.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.87 - 9.87i)T - 83iT^{2} \)
89 \( 1 + (2.07 - 7.76i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (14.2 + 3.82i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.87670079693909030295746967097, −9.626863972195689409732795298498, −8.284235967548683900337029244672, −7.55010883347419455457225250107, −6.97973614565388930694389608917, −6.36774159996397292865133162133, −5.19446906916552882821551520247, −3.11382631472821032878827262575, −2.64765597961651953418134244818, −1.25298645073371861075971639485, 1.54027957187770361049003990263, 3.22121541133903370730522119727, 4.19962185851977331236751983488, 5.43325515793321250871136409570, 5.71783757038694600099828564399, 7.13867676341677923601798095474, 8.312402834436818169312200835388, 9.268086480392125234137295225016, 9.794763652915189508253632045956, 10.77632692782488700756232445030

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