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Properties

Label 2-637-91.4-c1-0-21
Degree $2$
Conductor $637$
Sign $-0.144 - 0.989i$
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  + 2.70i·2-s + (0.172 + 0.299i)3-s − 5.30·4-s + (2.82 − 1.62i)5-s + (−0.809 + 0.467i)6-s − 8.94i·8-s + (1.44 − 2.49i)9-s + (4.40 + 7.62i)10-s + (1.59 − 0.923i)11-s + (−0.918 − 1.59i)12-s + (3.60 − 0.0186i)13-s + (0.976 + 0.563i)15-s + 13.5·16-s − 2.15·17-s + (6.74 + 3.89i)18-s + (2.07 + 1.20i)19-s + ⋯
L(s)  = 1  + 1.91i·2-s + (0.0998 + 0.172i)3-s − 2.65·4-s + (1.26 − 0.728i)5-s + (−0.330 + 0.190i)6-s − 3.16i·8-s + (0.480 − 0.831i)9-s + (1.39 + 2.41i)10-s + (0.482 − 0.278i)11-s + (−0.265 − 0.459i)12-s + (0.999 − 0.00517i)13-s + (0.252 + 0.145i)15-s + 3.38·16-s − 0.522·17-s + (1.58 + 0.917i)18-s + (0.477 + 0.275i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12060 + 1.29578i\)
\(L(\frac12)\) \(\approx\) \(1.12060 + 1.29578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0186i)T \)
good2 \( 1 - 2.70iT - 2T^{2} \)
3 \( 1 + (-0.172 - 0.299i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.82 + 1.62i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.59 + 0.923i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 + (-2.07 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.81T + 23T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.50 + 0.871i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.93iT - 37T^{2} \)
41 \( 1 + (-3.65 - 2.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.34 + 7.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.09 - 2.93i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.65 + 8.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (5.05 - 8.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.716 - 0.413i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.03 - 1.17i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.400 + 0.694i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (-7.99 + 4.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21003040314483108956666013324, −9.526102510795172313943118744907, −8.900893140882560696398777256860, −8.311348613420673824852853339229, −7.03490287163814326037319962673, −6.22008879153454507701148950488, −5.74946718872913947601895864941, −4.68304869080899919113580328226, −3.73803789701050844669013706306, −1.16892561846966301288255666567, 1.49106319130641087871097005855, 2.21784842121203748299058119153, 3.27454191755249299511368993266, 4.43396504537095990219870037260, 5.49457183944133694426096008039, 6.66286234449426205344093885627, 8.067879262775457537022487135208, 9.174807196193005215390257539259, 9.666102530741930797188385673972, 10.67313533622897818085302400843

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