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Properties

Label 2-637-49.37-c1-0-45
Degree $2$
Conductor $637$
Sign $0.266 + 0.963i$
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  + (1.47 − 1.00i)2-s + (1.63 − 1.51i)3-s + (0.440 − 1.12i)4-s + (1.13 + 0.349i)5-s + (0.889 − 3.89i)6-s + (0.173 + 2.64i)7-s + (0.316 + 1.38i)8-s + (0.148 − 1.98i)9-s + (2.02 − 0.625i)10-s + (−0.460 − 6.14i)11-s + (−0.983 − 2.50i)12-s + (−0.900 − 0.433i)13-s + (2.91 + 3.73i)14-s + (2.38 − 1.14i)15-s + (3.63 + 3.37i)16-s + (−4.08 − 0.615i)17-s + ⋯
L(s)  = 1  + (1.04 − 0.713i)2-s + (0.945 − 0.876i)3-s + (0.220 − 0.561i)4-s + (0.506 + 0.156i)5-s + (0.363 − 1.59i)6-s + (0.0656 + 0.997i)7-s + (0.111 + 0.490i)8-s + (0.0494 − 0.660i)9-s + (0.641 − 0.197i)10-s + (−0.138 − 1.85i)11-s + (−0.283 − 0.723i)12-s + (−0.249 − 0.120i)13-s + (0.780 + 0.996i)14-s + (0.615 − 0.296i)15-s + (0.908 + 0.843i)16-s + (−0.990 − 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.82997 - 2.15261i\)
\(L(\frac12)\) \(\approx\) \(2.82997 - 2.15261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.173 - 2.64i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (-1.47 + 1.00i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.63 + 1.51i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-1.13 - 0.349i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.460 + 6.14i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (4.08 + 0.615i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-2.80 + 4.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.54 + 0.384i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (5.42 - 6.79i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-3.62 - 6.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.57 + 4.00i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.67 - 7.34i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.00 - 4.41i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (3.87 - 2.64i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (0.482 - 1.23i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (12.0 - 3.70i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.86 + 4.76i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (4.08 + 7.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.72 - 7.18i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.03 - 0.708i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.83 + 6.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.20 - 2.50i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.00 - 13.3i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 5.42T + 97T^{2} \)
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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

−10.94304005083771402559346851628, −9.279883203361156021380522609591, −8.671134731833963860575487741255, −7.941366329457114812031744104975, −6.65789420239326107855175862894, −5.69692387107088027769397899025, −4.83706879023324739552444089999, −3.02209382307292690786003224342, −2.93044463960617351157159516332, −1.69970575117966539594373618142, 2.03193469950054778756870527306, 3.64051081319656249151324665001, 4.29114397530830527410622608797, 4.95337696630908200814060549337, 6.16382881736834955407288837559, 7.23011593373663882464139271631, 7.81904547333970281519729639824, 9.295506665895490629327969076451, 9.848999088462559843848689170260, 10.31359891645024716066016515622

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