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Properties

Label 2-637-91.80-c1-0-13
Degree $2$
Conductor $637$
Sign $0.0642 + 0.997i$
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  + (−0.521 − 1.94i)2-s − 1.44i·3-s + (−1.78 + 1.03i)4-s + (−0.849 + 3.16i)5-s + (−2.81 + 0.753i)6-s + (0.0872 + 0.0872i)8-s + 0.915·9-s + 6.61·10-s + (4.20 + 4.20i)11-s + (1.48 + 2.57i)12-s + (2.81 − 2.25i)13-s + (4.57 + 1.22i)15-s + (−1.93 + 3.35i)16-s + (0.314 + 0.544i)17-s + (−0.477 − 1.78i)18-s + (−0.521 − 0.521i)19-s + ⋯
L(s)  = 1  + (−0.368 − 1.37i)2-s − 0.833i·3-s + (−0.892 + 0.515i)4-s + (−0.379 + 1.41i)5-s + (−1.14 + 0.307i)6-s + (0.0308 + 0.0308i)8-s + 0.305·9-s + 2.09·10-s + (1.26 + 1.26i)11-s + (0.429 + 0.743i)12-s + (0.779 − 0.626i)13-s + (1.18 + 0.316i)15-s + (−0.484 + 0.838i)16-s + (0.0762 + 0.132i)17-s + (−0.112 − 0.420i)18-s + (−0.119 − 0.119i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0642 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0642 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0642 + 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0642 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923057 - 0.865530i\)
\(L(\frac12)\) \(\approx\) \(0.923057 - 0.865530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.81 + 2.25i)T \)
good2 \( 1 + (0.521 + 1.94i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 1.44iT - 3T^{2} \)
5 \( 1 + (0.849 - 3.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.20 - 4.20i)T + 11iT^{2} \)
17 \( 1 + (-0.314 - 0.544i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.521 + 0.521i)T + 19iT^{2} \)
23 \( 1 + (-3.93 - 2.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.33 + 2.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.06 + 0.285i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.44 - 0.655i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.746 - 2.78i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.49 - 2.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.10 - 1.90i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.89 + 6.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.919 + 0.246i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 1.08iT - 61T^{2} \)
67 \( 1 + (3.81 - 3.81i)T - 67iT^{2} \)
71 \( 1 + (-0.590 - 2.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.59 - 5.94i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.08 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.59 - 3.59i)T + 83iT^{2} \)
89 \( 1 + (1.05 + 3.94i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.41 + 2.25i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.44550490498187159356508448229, −9.857308976676216571902484834403, −8.877593521682879476937482291005, −7.61149439719623439062220959160, −6.91905357598937059095425993419, −6.23508995121405682642370475016, −4.21919451601443896884220524821, −3.36900236340059017372675947526, −2.27253349708970913243378670111, −1.22486517351914689764618465408, 1.02511237451973929791483396223, 3.67124154334806223913962439978, 4.49130382351582480611079927131, 5.40885760298838648277069944489, 6.30101118048614919553385019231, 7.26833793377801957201065437805, 8.435455989476137466166709056878, 9.004874666939478456093706077516, 9.203065892125411487758839173013, 10.66704666195236596400125254017

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