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Properties

Label 2-637-91.83-c1-0-9
Degree $2$
Conductor $637$
Sign $0.418 - 0.908i$
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  + (−1.74 + 1.74i)2-s − 2.04i·3-s − 4.11i·4-s + (1.15 + 1.15i)5-s + (3.57 + 3.57i)6-s + (3.69 + 3.69i)8-s − 1.17·9-s − 4.04·10-s + (4.06 + 4.06i)11-s − 8.41·12-s + (−3.57 + 0.473i)13-s + (2.36 − 2.36i)15-s − 4.69·16-s − 1.98·17-s + (2.06 − 2.06i)18-s + (0.672 + 0.672i)19-s + ⋯
L(s)  = 1  + (−1.23 + 1.23i)2-s − 1.18i·3-s − 2.05i·4-s + (0.517 + 0.517i)5-s + (1.45 + 1.45i)6-s + (1.30 + 1.30i)8-s − 0.393·9-s − 1.28·10-s + (1.22 + 1.22i)11-s − 2.42·12-s + (−0.991 + 0.131i)13-s + (0.611 − 0.611i)15-s − 1.17·16-s − 0.481·17-s + (0.485 − 0.485i)18-s + (0.154 + 0.154i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724703 + 0.464224i\)
\(L(\frac12)\) \(\approx\) \(0.724703 + 0.464224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.473i)T \)
good2 \( 1 + (1.74 - 1.74i)T - 2iT^{2} \)
3 \( 1 + 2.04iT - 3T^{2} \)
5 \( 1 + (-1.15 - 1.15i)T + 5iT^{2} \)
11 \( 1 + (-4.06 - 4.06i)T + 11iT^{2} \)
17 \( 1 + 1.98T + 17T^{2} \)
19 \( 1 + (-0.672 - 0.672i)T + 19iT^{2} \)
23 \( 1 - 3.54iT - 23T^{2} \)
29 \( 1 - 2.83T + 29T^{2} \)
31 \( 1 + (-3.17 - 3.17i)T + 31iT^{2} \)
37 \( 1 + (-2.73 - 2.73i)T + 37iT^{2} \)
41 \( 1 + (-4.02 - 4.02i)T + 41iT^{2} \)
43 \( 1 - 5.30iT - 43T^{2} \)
47 \( 1 + (-0.328 + 0.328i)T - 47iT^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (-8.48 + 8.48i)T - 59iT^{2} \)
61 \( 1 - 5.05iT - 61T^{2} \)
67 \( 1 + (4.29 - 4.29i)T - 67iT^{2} \)
71 \( 1 + (4.84 - 4.84i)T - 71iT^{2} \)
73 \( 1 + (-3.10 + 3.10i)T - 73iT^{2} \)
79 \( 1 - 6.16T + 79T^{2} \)
83 \( 1 + (11.5 + 11.5i)T + 83iT^{2} \)
89 \( 1 + (2.57 - 2.57i)T - 89iT^{2} \)
97 \( 1 + (-7.09 - 7.09i)T + 97iT^{2} \)
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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

−10.11713794625114293347233871672, −9.803088810662228151480487768828, −8.854476301575965081868499746515, −7.84956798221398287752030684958, −7.08157800674026025444881752569, −6.73645215396667368561815133840, −5.97704920355898484565322009172, −4.58910002604627735410556533050, −2.29339931942176414672425673301, −1.22746448956740456018045415418, 0.850752601260156896186628995441, 2.38432083827330513144280570683, 3.57407318927833557825034546012, 4.45769082191824968136216055786, 5.72482251487184904404364763982, 7.16789454263632574043055064407, 8.533659201661712021587624972361, 8.993632582772886744933373674701, 9.616474433301787630923359577692, 10.27342268887258578823044717197

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