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Properties

Label 2-637-7.2-c1-0-17
Degree $2$
Conductor $637$
Sign $0.701 + 0.712i$
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 − 1.73i)2-s + (−0.999 − 1.73i)4-s + (−1.5 + 2.59i)5-s + (1.5 − 2.59i)9-s + (3 + 5.19i)10-s + (3 + 5.19i)11-s + 13-s + (1.99 − 3.46i)16-s + (2 + 3.46i)17-s + (−3 − 5.19i)18-s + (2.5 − 4.33i)19-s + 5.99·20-s + 12·22-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + (1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (−0.670 + 1.16i)5-s + (0.5 − 0.866i)9-s + (0.948 + 1.64i)10-s + (0.904 + 1.56i)11-s + 0.277·13-s + (0.499 − 0.866i)16-s + (0.485 + 0.840i)17-s + (−0.707 − 1.22i)18-s + (0.573 − 0.993i)19-s + 1.34·20-s + 2.55·22-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01572 - 0.844757i\)
\(L(\frac12)\) \(\approx\) \(2.01572 - 0.844757i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.65692092460855196745324440424, −9.913449355444814881784477685814, −9.167391887771824174634402017319, −7.35196007869106240365006618850, −7.18908742508590601313968260101, −5.81999417995825630018858231590, −4.20341065449296861761035057529, −3.90337222200630746688036814772, −2.80261427554416638192209594583, −1.50938155335232479980636395902, 1.20919581381145784545195474837, 3.57050520560488258897162781027, 4.39517148428385295574554083289, 5.31296843448578688893409175778, 5.99197996786868933247458472617, 7.15366505102637705856941077257, 8.032324588200659913296266724006, 8.470982317285759574967752985939, 9.568364925883983481630577943600, 10.85136710588636109608076253036

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