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Properties

Label 2-637-13.3-c1-0-19
Degree $2$
Conductor $637$
Sign $-0.859 - 0.511i$
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  + (1.20 + 2.09i)2-s + (0.707 + 1.22i)3-s + (−1.91 + 3.31i)4-s + 3.82·5-s + (−1.70 + 2.95i)6-s − 4.41·8-s + (0.500 − 0.866i)9-s + (4.62 + 8.00i)10-s + (−1.70 − 2.95i)11-s − 5.41·12-s + (−3.5 + 0.866i)13-s + (2.70 + 4.68i)15-s + (−1.49 − 2.59i)16-s + (0.0857 − 0.148i)17-s + 2.41·18-s + (−3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (0.408 + 0.707i)3-s + (−0.957 + 1.65i)4-s + 1.71·5-s + (−0.696 + 1.20i)6-s − 1.56·8-s + (0.166 − 0.288i)9-s + (1.46 + 2.53i)10-s + (−0.514 − 0.891i)11-s − 1.56·12-s + (−0.970 + 0.240i)13-s + (0.698 + 1.21i)15-s + (−0.374 − 0.649i)16-s + (0.0208 − 0.0360i)17-s + 0.569·18-s + (−0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.859 - 0.511i$
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: \((2,\ 637,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.825761 + 3.00458i\)
\(L(\frac12)\) \(\approx\) \(0.825761 + 3.00458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.82T + 5T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.91 + 8.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.41T + 31T^{2} \)
37 \( 1 + (3.74 + 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.91 + 5.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.292 - 0.507i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.65T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-0.878 + 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.91 - 8.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.12 - 3.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.171 + 0.297i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.656T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-3.65 - 6.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.58 + 4.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53537344736185325739418189817, −9.911842058555414069509275326960, −9.074387541133682533308569607010, −8.274674383645498440734208224663, −7.16419553915918597119040530313, −6.10673235977313205134280288605, −5.73087250125819981732741227793, −4.72048933824831001552265034074, −3.72873554234885787896417859087, −2.36355300339078329178547703560, 1.54281582164676335827024476909, 2.24999537054395255814031732844, 2.92011694089387734672592693340, 4.80169613690520254540019697438, 5.13388686821245546767550560501, 6.47814749218052134902352559733, 7.43260939826399787141804643430, 8.874020750485172709763090732941, 9.793463886654685386237126329963, 10.25558542704865597248181052229

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