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Properties

Label 2-637-91.74-c1-0-11
Degree $2$
Conductor $637$
Sign $-0.150 - 0.988i$
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-s + (0.707 + 1.22i)3-s − 4-s + (2.04 + 3.54i)5-s + (0.707 + 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (2.04 + 3.54i)10-s + (1.89 + 3.28i)11-s + (−0.707 − 1.22i)12-s + (0.634 − 3.54i)13-s + (−2.89 + 5.01i)15-s − 16-s + 1.26·17-s + (0.500 − 0.866i)18-s + (−1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.707i)3-s − 0.5·4-s + (0.916 + 1.58i)5-s + (0.288 + 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (0.647 + 1.12i)10-s + (0.572 + 0.991i)11-s + (−0.204 − 0.353i)12-s + (0.176 − 0.984i)13-s + (−0.748 + 1.29i)15-s − 0.250·16-s + 0.307·17-s + (0.117 − 0.204i)18-s + (−0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50774 + 1.75451i\)
\(L(\frac12)\) \(\approx\) \(1.50774 + 1.75451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.634 + 3.54i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.04 - 3.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 + (0.397 - 0.689i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.79T + 37T^{2} \)
41 \( 1 + (-1.48 + 2.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.29 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + (-4.17 + 7.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.89 - 3.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.29 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.10 - 1.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (2.12 + 3.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40430753264648144294814926925, −10.02640775791520888055423410703, −9.451578263986042712962960644513, −8.295262877842338453086732786552, −7.01219247005732377602147540802, −6.17964486071976838736349510736, −5.35549836660147707825883354497, −3.97650937219504191883003559191, −3.46156387207036421448766120085, −2.26018287278257359617212917489, 1.06033177312534432396940287417, 2.27731004662655525782284625794, 3.97754157432962080753699380487, 4.73740050865109934858410520267, 5.75591523060618035635753023824, 6.39657867040960746355440494519, 7.957187442760346795507823931267, 8.711878898684621741259535400770, 9.188353691690544488719688867250, 10.09677256769421484049266856797

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