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Properties

Label 2-637-13.10-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.874 + 0.484i$
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  + (−2.34 + 1.35i)2-s + (−0.172 − 0.299i)3-s + (2.65 − 4.59i)4-s + 3.25i·5-s + (0.809 + 0.467i)6-s + 8.94i·8-s + (1.44 − 2.49i)9-s + (−4.40 − 7.62i)10-s + (−1.59 + 0.923i)11-s − 1.83·12-s + (−3.60 − 0.0186i)13-s + (0.976 − 0.563i)15-s + (−6.77 − 11.7i)16-s + (−1.07 + 1.86i)17-s + 7.78i·18-s + (2.07 + 1.20i)19-s + ⋯
L(s)  = 1  + (−1.65 + 0.955i)2-s + (−0.0998 − 0.172i)3-s + (1.32 − 2.29i)4-s + 1.45i·5-s + (0.330 + 0.190i)6-s + 3.16i·8-s + (0.480 − 0.831i)9-s + (−1.39 − 2.41i)10-s + (−0.482 + 0.278i)11-s − 0.530·12-s + (−0.999 − 0.00517i)13-s + (0.252 − 0.145i)15-s + (−1.69 − 2.93i)16-s + (−0.261 + 0.452i)17-s + 1.83i·18-s + (0.477 + 0.275i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0651223 - 0.252091i\)
\(L(\frac12)\) \(\approx\) \(0.0651223 - 0.252091i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 0.0186i)T \)
good2 \( 1 + (2.34 - 1.35i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.172 + 0.299i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.25iT - 5T^{2} \)
11 \( 1 + (1.59 - 0.923i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.07 - 1.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.07 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.906 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (5.14 - 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.65 - 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.34 - 7.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.87iT - 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 + (9.31 + 5.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.05 + 8.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.716 + 0.413i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.03 + 1.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.19iT - 73T^{2} \)
79 \( 1 - 0.801T + 79T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + (13.0 - 7.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.99 + 4.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69119104332016837387309183958, −9.938619361701919234115014695011, −9.546740400207982405016672387512, −8.288212492605918666207084718710, −7.47893302178708143544916633819, −6.82441021527843382342360418601, −6.36827125235760301784582326700, −5.12627087211969657961939697512, −3.09388419833914978482892697174, −1.71710855028552178484169654058, 0.24256294292656162662769073894, 1.62157684620391060012955346171, 2.74766018751489197640235894083, 4.32928071752310086518766831327, 5.27338116449209913399331136091, 7.09338646485165694984853286128, 7.81918401322865797144794058941, 8.577419949422989868532608255815, 9.283803532746185914944455890365, 9.987857909489025776790892431225

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