Properties

Label 2-637-91.30-c1-0-37
Degree $2$
Conductor $637$
Sign $0.489 + 0.872i$
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  + (1.99 − 1.15i)2-s + 1.47·3-s + (1.65 − 2.86i)4-s + (0.733 + 0.423i)5-s + (2.93 − 1.69i)6-s − 3.00i·8-s − 0.829·9-s + 1.95·10-s + 1.50i·11-s + (2.43 − 4.21i)12-s + (2.92 − 2.11i)13-s + (1.08 + 0.624i)15-s + (−0.156 − 0.271i)16-s + (−1.03 + 1.79i)17-s + (−1.65 + 0.954i)18-s + 0.0474i·19-s + ⋯
L(s)  = 1  + (1.41 − 0.814i)2-s + 0.850·3-s + (0.826 − 1.43i)4-s + (0.328 + 0.189i)5-s + (1.19 − 0.692i)6-s − 1.06i·8-s − 0.276·9-s + 0.617·10-s + 0.453i·11-s + (0.702 − 1.21i)12-s + (0.810 − 0.585i)13-s + (0.279 + 0.161i)15-s + (−0.0391 − 0.0678i)16-s + (−0.251 + 0.435i)17-s + (−0.389 + 0.225i)18-s + 0.0108i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.489 + 0.872i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.489 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.52966 - 2.06666i\)
\(L(\frac12)\) \(\approx\) \(3.52966 - 2.06666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 + 2.11i)T \)
good2 \( 1 + (-1.99 + 1.15i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 1.47T + 3T^{2} \)
5 \( 1 + (-0.733 - 0.423i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.50iT - 11T^{2} \)
17 \( 1 + (1.03 - 1.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 0.0474iT - 19T^{2} \)
23 \( 1 + (3.90 + 6.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.679 - 1.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.80 - 3.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.80 - 3.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.67 - 5.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.311 + 0.180i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.35 + 2.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.42 + 0.820i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + 2.04iT - 67T^{2} \)
71 \( 1 + (-12.3 + 7.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.85 + 3.38i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.82 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + (15.1 - 8.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.369 - 0.213i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.69734480038370661282327266406, −9.830668659879024252226377037630, −8.693982309667533447032028605954, −7.929816749663103806847771932404, −6.43614583413983625937072760564, −5.75746630377570822091429698014, −4.57448041779656612550755849878, −3.64778441179956826118987749031, −2.76581210648882862981858299235, −1.85584206817488073364070633813, 2.13700246779837290931008583285, 3.49109951150249281240096978747, 4.01440858535582363523018848983, 5.53868012177169431872765585803, 5.84004120957120506045160812120, 7.13287297263994110087840041347, 7.81858703591928334181270003821, 8.913641000712505107625627153280, 9.500854218614467968459770141450, 11.05106173118017253300077277238

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