Properties

Label 2-637-91.33-c1-0-31
Degree $2$
Conductor $637$
Sign $-0.548 + 0.836i$
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  + (0.554 − 2.06i)2-s + 0.0197i·3-s + (−2.24 − 1.29i)4-s + (0.360 + 1.34i)5-s + (0.0408 + 0.0109i)6-s + (−0.893 + 0.893i)8-s + 2.99·9-s + 2.98·10-s + (−0.246 + 0.246i)11-s + (0.0255 − 0.0442i)12-s + (−1.32 − 3.35i)13-s + (−0.0265 + 0.00711i)15-s + (−1.23 − 2.14i)16-s + (0.491 − 0.850i)17-s + (1.66 − 6.20i)18-s + (3.25 − 3.25i)19-s + ⋯
L(s)  = 1  + (0.392 − 1.46i)2-s + 0.0113i·3-s + (−1.12 − 0.647i)4-s + (0.161 + 0.601i)5-s + (0.0166 + 0.00446i)6-s + (−0.315 + 0.315i)8-s + 0.999·9-s + 0.943·10-s + (−0.0743 + 0.0743i)11-s + (0.00737 − 0.0127i)12-s + (−0.368 − 0.929i)13-s + (−0.00685 + 0.00183i)15-s + (−0.309 − 0.535i)16-s + (0.119 − 0.206i)17-s + (0.392 − 1.46i)18-s + (0.747 − 0.747i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.548 + 0.836i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.548 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924395 - 1.71209i\)
\(L(\frac12)\) \(\approx\) \(0.924395 - 1.71209i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 3.35i)T \)
good2 \( 1 + (-0.554 + 2.06i)T + (-1.73 - i)T^{2} \)
3 \( 1 - 0.0197iT - 3T^{2} \)
5 \( 1 + (-0.360 - 1.34i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.246 - 0.246i)T - 11iT^{2} \)
17 \( 1 + (-0.491 + 0.850i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.25 + 3.25i)T - 19iT^{2} \)
23 \( 1 + (-2.86 + 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.941 - 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.81 - 0.755i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.91 + 2.12i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.580 - 2.16i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.47 - 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.5 + 2.83i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.77 - 6.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (14.7 - 3.94i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 + (-7.13 - 7.13i)T + 67iT^{2} \)
71 \( 1 + (-2.43 + 9.07i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.76 - 10.3i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.890 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.33 + 8.33i)T - 83iT^{2} \)
89 \( 1 + (3.51 - 13.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.6 + 3.39i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.45996031287226031879054989340, −9.865625441232546429543322589343, −8.941574896045652390747774429903, −7.49149627106687372271358480172, −6.83172061563693093876453194549, −5.30675488063260539359080209947, −4.46905281303080451257106246237, −3.27087310423374843317713895754, −2.53489207864306739551406801745, −1.08736122411522090770657638511, 1.65653089057888374275232569904, 3.74681335819942387628584874835, 4.76643021732333730849648484265, 5.39359041796029878932709231509, 6.53095166751474506384689171477, 7.19614744498124585990439445670, 7.995703891212363047280485794612, 8.955142958197618984110801911217, 9.700418643137504919041144991547, 10.77608129397622435971805502814

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