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Properties

Label 2-637-91.80-c1-0-11
Degree $2$
Conductor $637$
Sign $-0.135 - 0.990i$
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  + (0.664 + 2.47i)2-s − 2.69i·3-s + (−3.97 + 2.29i)4-s + (−0.103 + 0.384i)5-s + (6.68 − 1.79i)6-s + (−4.70 − 4.70i)8-s − 4.27·9-s − 1.02·10-s + (2.56 + 2.56i)11-s + (6.19 + 10.7i)12-s + (3.44 + 1.06i)13-s + (1.03 + 0.277i)15-s + (3.95 − 6.84i)16-s + (2.04 + 3.54i)17-s + (−2.83 − 10.5i)18-s + (−0.569 − 0.569i)19-s + ⋯
L(s)  = 1  + (0.469 + 1.75i)2-s − 1.55i·3-s + (−1.98 + 1.14i)4-s + (−0.0461 + 0.172i)5-s + (2.72 − 0.731i)6-s + (−1.66 − 1.66i)8-s − 1.42·9-s − 0.323·10-s + (0.774 + 0.774i)11-s + (1.78 + 3.09i)12-s + (0.955 + 0.294i)13-s + (0.267 + 0.0717i)15-s + (0.987 − 1.71i)16-s + (0.496 + 0.860i)17-s + (−0.668 − 2.49i)18-s + (−0.130 − 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.135 - 0.990i$
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: \((2,\ 637,\ (\ :1/2),\ -0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13665 + 1.30220i\)
\(L(\frac12)\) \(\approx\) \(1.13665 + 1.30220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.44 - 1.06i)T \)
good2 \( 1 + (-0.664 - 2.47i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 2.69iT - 3T^{2} \)
5 \( 1 + (0.103 - 0.384i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.56 - 2.56i)T + 11iT^{2} \)
17 \( 1 + (-2.04 - 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.569 + 0.569i)T + 19iT^{2} \)
23 \( 1 + (-4.41 - 2.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.06 + 1.62i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.73 - 0.463i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.578 + 2.15i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.19 + 2.19i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.17 + 1.92i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.77iT - 61T^{2} \)
67 \( 1 + (-3.55 + 3.55i)T - 67iT^{2} \)
71 \( 1 + (0.582 + 2.17i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.28 + 4.78i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.80 - 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.36 - 5.36i)T + 83iT^{2} \)
89 \( 1 + (-3.09 - 11.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.71 - 2.06i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.01904665005407054075362559861, −9.424679052162582052902302537510, −8.528987425354142248443966326504, −7.896356201577109854829078439174, −6.96716699771322717732483625337, −6.62012396256252407122942185113, −5.84299621366624609347032270501, −4.67426979775505984756993907775, −3.41175055253690682899710915447, −1.43726274073316273547790007095, 0.994437976917343482999915804185, 2.95935652522622003441419112400, 3.50122622202251584714897121907, 4.50964464555223953475089262646, 5.09516260424555306669813451974, 6.27923405382957996306404814108, 8.466158001421404519348023379362, 9.003811730840921770307512885727, 9.839218660713167752273334945158, 10.43951936518092533813367664410

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