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Properties

Label 2-637-13.4-c1-0-39
Degree $2$
Conductor $637$
Sign $-0.800 - 0.599i$
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  + (−1.16 − 0.672i)2-s + (1.02 − 1.77i)3-s + (−0.0951 − 0.164i)4-s − 3.56i·5-s + (−2.38 + 1.37i)6-s + 2.94i·8-s + (−0.601 − 1.04i)9-s + (−2.39 + 4.15i)10-s + (1.10 + 0.639i)11-s − 0.390·12-s + (−3.57 − 0.474i)13-s + (−6.33 − 3.65i)15-s + (1.79 − 3.10i)16-s + (−3.86 − 6.70i)17-s + 1.61i·18-s + (0.817 − 0.471i)19-s + ⋯
L(s)  = 1  + (−0.823 − 0.475i)2-s + (0.591 − 1.02i)3-s + (−0.0475 − 0.0824i)4-s − 1.59i·5-s + (−0.975 + 0.562i)6-s + 1.04i·8-s + (−0.200 − 0.347i)9-s + (−0.758 + 1.31i)10-s + (0.333 + 0.192i)11-s − 0.112·12-s + (−0.991 − 0.131i)13-s + (−1.63 − 0.944i)15-s + (0.447 − 0.775i)16-s + (−0.938 − 1.62i)17-s + 0.381i·18-s + (0.187 − 0.108i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266575 + 0.800725i\)
\(L(\frac12)\) \(\approx\) \(0.266575 + 0.800725i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 + 0.474i)T \)
good2 \( 1 + (1.16 + 0.672i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
11 \( 1 + (-1.10 - 0.639i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.86 + 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.817 + 0.471i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.823 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 - 3.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.15iT - 31T^{2} \)
37 \( 1 + (0.914 + 0.528i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.63 - 2.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 - 3.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.894iT - 47T^{2} \)
53 \( 1 + 0.0799T + 53T^{2} \)
59 \( 1 + (-9.68 + 5.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.81 + 6.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.47 + 3.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.89 + 5.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.760iT - 73T^{2} \)
79 \( 1 + 2.85T + 79T^{2} \)
83 \( 1 - 2.32iT - 83T^{2} \)
89 \( 1 + (-6.56 - 3.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.414 + 0.239i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.606960668301299009852835825144, −9.202809755167502797066467317120, −8.493300559387558637800462121605, −7.71685303620538301322969607416, −6.82907079546027855085354334086, −5.20040787649537165688855367655, −4.70540720840052067700875476305, −2.63016568604490413660556673652, −1.63466827974243403840762533451, −0.56897358995766931228504353134, 2.47153750208147208769271121449, 3.64395659655235339850420020459, 4.20483024781195384943523792303, 6.03832462367957708617179408061, 6.90154053463817880902039140493, 7.64605666025789201412494127180, 8.616588849375296754544251532396, 9.392087138813841457141135130408, 10.10147786955362918193258749472, 10.61810297611562419015974553129

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