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Properties

Label 2-637-13.3-c1-0-31
Degree $2$
Conductor $637$
Sign $0.281 + 0.959i$
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  + (0.134 + 0.232i)2-s + (0.571 + 0.989i)3-s + (0.964 − 1.66i)4-s − 2.56·5-s + (−0.153 + 0.265i)6-s + 1.05·8-s + (0.846 − 1.46i)9-s + (−0.343 − 0.594i)10-s + (−1.97 − 3.41i)11-s + 2.20·12-s + (−3.15 − 1.74i)13-s + (−1.46 − 2.53i)15-s + (−1.78 − 3.09i)16-s + (−0.392 + 0.679i)17-s + 0.454·18-s + (3.74 − 6.49i)19-s + ⋯
L(s)  = 1  + (0.0947 + 0.164i)2-s + (0.329 + 0.571i)3-s + (0.482 − 0.834i)4-s − 1.14·5-s + (−0.0625 + 0.108i)6-s + 0.372·8-s + (0.282 − 0.488i)9-s + (−0.108 − 0.188i)10-s + (−0.594 − 1.03i)11-s + 0.636·12-s + (−0.874 − 0.484i)13-s + (−0.378 − 0.654i)15-s + (−0.446 − 0.773i)16-s + (−0.0952 + 0.164i)17-s + 0.107·18-s + (0.859 − 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07276 - 0.802886i\)
\(L(\frac12)\) \(\approx\) \(1.07276 - 0.802886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 + 1.74i)T \)
good2 \( 1 + (-0.134 - 0.232i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.571 - 0.989i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
11 \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.392 - 0.679i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 + 6.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.97 - 6.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.17 + 2.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 + (3.37 + 5.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.21 - 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 - 9.27T + 53T^{2} \)
59 \( 1 + (-4.48 + 7.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.72 - 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.768T + 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 + (3.83 + 6.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 + 2.05i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.43987396738645960755460201715, −9.550430109277602844138019933427, −8.790756744976507445952713060216, −7.52164330670314438368212668847, −7.10410100987095381520116331331, −5.67561051926466589681581648015, −4.92701377484401482720796342877, −3.72065583743998813947118504416, −2.77443250128593687561481278172, −0.67176075289715635544913054514, 1.91465519376068749623184461451, 2.93681885303542192752732179998, 4.12055322569979237837222866411, 4.96739541670003912933788044764, 6.78957812018491143009843591386, 7.45880398169572080421942994470, 7.79837898838485130080248198842, 8.709473688446654480274541762303, 10.06184411355804235076940243072, 10.85322422558220201522926752692

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