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Properties

Label 2-637-91.83-c1-0-32
Degree $2$
Conductor $637$
Sign $-0.181 + 0.983i$
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  + (−0.508 + 0.508i)2-s − 1.66i·3-s + 1.48i·4-s + (−1.36 − 1.36i)5-s + (0.846 + 0.846i)6-s + (−1.77 − 1.77i)8-s + 0.230·9-s + 1.39·10-s + (2.25 + 2.25i)11-s + 2.46·12-s + (−0.846 − 3.50i)13-s + (−2.27 + 2.27i)15-s − 1.16·16-s − 0.508·17-s + (−0.116 + 0.116i)18-s + (−1.94 − 1.94i)19-s + ⋯
L(s)  = 1  + (−0.359 + 0.359i)2-s − 0.960i·3-s + 0.741i·4-s + (−0.612 − 0.612i)5-s + (0.345 + 0.345i)6-s + (−0.626 − 0.626i)8-s + 0.0766·9-s + 0.440·10-s + (0.680 + 0.680i)11-s + 0.712·12-s + (−0.234 − 0.972i)13-s + (−0.588 + 0.588i)15-s − 0.291·16-s − 0.123·17-s + (−0.0275 + 0.0275i)18-s + (−0.445 − 0.445i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.181 + 0.983i$
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: \((2,\ 637,\ (\ :1/2),\ -0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515676 - 0.619770i\)
\(L(\frac12)\) \(\approx\) \(0.515676 - 0.619770i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.846 + 3.50i)T \)
good2 \( 1 + (0.508 - 0.508i)T - 2iT^{2} \)
3 \( 1 + 1.66iT - 3T^{2} \)
5 \( 1 + (1.36 + 1.36i)T + 5iT^{2} \)
11 \( 1 + (-2.25 - 2.25i)T + 11iT^{2} \)
17 \( 1 + 0.508T + 17T^{2} \)
19 \( 1 + (1.94 + 1.94i)T + 19iT^{2} \)
23 \( 1 + 2.88iT - 23T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + (2.26 + 2.26i)T + 31iT^{2} \)
37 \( 1 + (6.88 + 6.88i)T + 37iT^{2} \)
41 \( 1 + (-5.34 - 5.34i)T + 41iT^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 + (7.89 - 7.89i)T - 47iT^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 + (-2.74 + 2.74i)T - 59iT^{2} \)
61 \( 1 + 6.37iT - 61T^{2} \)
67 \( 1 + (4.80 - 4.80i)T - 67iT^{2} \)
71 \( 1 + (1.90 - 1.90i)T - 71iT^{2} \)
73 \( 1 + (-0.184 + 0.184i)T - 73iT^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (5.86 + 5.86i)T + 83iT^{2} \)
89 \( 1 + (-8.66 + 8.66i)T - 89iT^{2} \)
97 \( 1 + (-7.04 - 7.04i)T + 97iT^{2} \)
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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

−10.22247984737594215086877468211, −9.118345893482059089410198725507, −8.425408130458862169337986591818, −7.59627930229642283295263430241, −7.11996150711083325680169329131, −6.17326669515682019874677723517, −4.66682288327814108511330188665, −3.72735441876260898763664835901, −2.21997882379374645563162277571, −0.50289823671559973060518391514, 1.64380831764169904536867234084, 3.28686657405708730166079599930, 4.16343782089095789906539150840, 5.22635959979918139161839204917, 6.35145775061057698972964090275, 7.22178057448281099584263435125, 8.594964398527794404042221325557, 9.293145645134671863641276123039, 9.997768018033771821032184164419, 10.78565882799157635013510874697

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