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Properties

Label 2-637-91.89-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.889 - 0.456i$
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  + (−0.270 − 0.270i)2-s + (−0.792 − 0.457i)3-s − 1.85i·4-s + (−0.959 − 3.58i)5-s + (0.0906 + 0.338i)6-s + (−1.04 + 1.04i)8-s + (−1.08 − 1.87i)9-s + (−0.709 + 1.22i)10-s + (0.0226 + 0.0846i)11-s + (−0.847 + 1.46i)12-s + (−1.63 − 3.21i)13-s + (−0.877 + 3.27i)15-s − 3.14·16-s + 5.89·17-s + (−0.214 + 0.799i)18-s + (3.58 + 0.960i)19-s + ⋯
L(s)  = 1  + (−0.191 − 0.191i)2-s + (−0.457 − 0.264i)3-s − 0.926i·4-s + (−0.429 − 1.60i)5-s + (0.0369 + 0.138i)6-s + (−0.368 + 0.368i)8-s + (−0.360 − 0.624i)9-s + (−0.224 + 0.388i)10-s + (0.00683 + 0.0255i)11-s + (−0.244 + 0.423i)12-s + (−0.453 − 0.891i)13-s + (−0.226 + 0.845i)15-s − 0.785·16-s + 1.42·17-s + (−0.0505 + 0.188i)18-s + (0.822 + 0.220i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180120 + 0.745266i\)
\(L(\frac12)\) \(\approx\) \(0.180120 + 0.745266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 3.21i)T \)
good2 \( 1 + (0.270 + 0.270i)T + 2iT^{2} \)
3 \( 1 + (0.792 + 0.457i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.959 + 3.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0226 - 0.0846i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 + (-3.58 - 0.960i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.446iT - 23T^{2} \)
29 \( 1 + (-0.706 - 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.94 - 0.520i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.87 + 1.87i)T - 37iT^{2} \)
41 \( 1 + (3.00 + 0.804i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.64 - 4.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.84 - 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.28 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.05 + 5.05i)T + 59iT^{2} \)
61 \( 1 + (0.110 - 0.0638i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.61 + 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.83 - 2.63i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.37 + 8.84i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.75 + 3.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.17 + 2.17i)T - 83iT^{2} \)
89 \( 1 + (1.19 + 1.19i)T + 89iT^{2} \)
97 \( 1 + (-0.452 - 1.68i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.850724116226166386814056990712, −9.420031956295657015871976545530, −8.395936744144732789569775620490, −7.61253090452230134581390107615, −6.19074447042835722593798439610, −5.40216554968610571365055697469, −4.86032161288860213288608208479, −3.31170839264567900774265418684, −1.35164300254024262848390557557, −0.51398356843318927520859061692, 2.58098980172235139529503332786, 3.36931673186291410345526717944, 4.49035746882987082669654060472, 5.83605137960232007298649640141, 6.85614790899922735958219379742, 7.51376753786637061324929024903, 8.165812204812510477216387852173, 9.485570447613898122544371125837, 10.27943336133326080915490173632, 11.20335048133791097769056939355

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