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Properties

Label 2-637-91.88-c1-0-1
Degree $2$
Conductor $637$
Sign $-0.997 + 0.0651i$
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  + (−1.82 − 1.05i)2-s − 2.26·3-s + (1.22 + 2.12i)4-s + (−3.11 + 1.80i)5-s + (4.13 + 2.38i)6-s − 0.948i·8-s + 2.11·9-s + 7.59·10-s + 0.886i·11-s + (−2.76 − 4.79i)12-s + (1.17 + 3.40i)13-s + (7.05 − 4.07i)15-s + (1.44 − 2.51i)16-s + (2.48 + 4.29i)17-s + (−3.86 − 2.23i)18-s + 2.37i·19-s + ⋯
L(s)  = 1  + (−1.29 − 0.745i)2-s − 1.30·3-s + (0.612 + 1.06i)4-s + (−1.39 + 0.805i)5-s + (1.68 + 0.973i)6-s − 0.335i·8-s + 0.705·9-s + 2.40·10-s + 0.267i·11-s + (−0.799 − 1.38i)12-s + (0.325 + 0.945i)13-s + (1.82 − 1.05i)15-s + (0.362 − 0.627i)16-s + (0.601 + 1.04i)17-s + (−0.910 − 0.525i)18-s + 0.545i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.997 + 0.0651i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.997 + 0.0651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00246577 - 0.0756326i\)
\(L(\frac12)\) \(\approx\) \(0.00246577 - 0.0756326i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 3.40i)T \)
good2 \( 1 + (1.82 + 1.05i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 2.26T + 3T^{2} \)
5 \( 1 + (3.11 - 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.886iT - 11T^{2} \)
17 \( 1 + (-2.48 - 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 2.37iT - 19T^{2} \)
23 \( 1 + (1.92 - 3.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.640 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.33 + 4.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.34 - 4.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.4 - 6.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.58 - 1.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.46 - 4.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.34 + 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.53T + 61T^{2} \)
67 \( 1 - 8.42iT - 67T^{2} \)
71 \( 1 + (5.58 + 3.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.19 + 3.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.656i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.76iT - 83T^{2} \)
89 \( 1 + (-3.13 - 1.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.401 - 0.231i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.06444085896569715027524844660, −10.36284467320555436517161857267, −9.604625354969676587774933778859, −8.331055150122133309890675107964, −7.72215616594144008015394793841, −6.78189858553128348003727091602, −5.79423336507909861301467446763, −4.33954736376678819303103048854, −3.28698326564594000462381065060, −1.55796398827128677551117613758, 0.11600703205403235142566166106, 0.841040484710832615602534992357, 3.60486671046944158580367125217, 4.94339718376539608044709208771, 5.67977847063941072045691647880, 6.84376642662151660644641017035, 7.53829874042115436121686931945, 8.363493855745299046790974996489, 8.994260137283482704101689011961, 10.15953112642225937518494405104

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