↑

Properties

Label 2-637-91.9-c1-0-0
Degree $2$
Conductor $637$
Sign $0.0690 + 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 2.37i)2-s − 1.36·3-s + (−2.75 + 4.77i)4-s + (0.370 − 0.641i)5-s + (−1.87 − 3.23i)6-s − 9.63·8-s − 1.13·9-s + 2.03·10-s − 1.36·11-s + (3.76 − 6.51i)12-s + (−0.301 − 3.59i)13-s + (−0.505 + 0.875i)15-s + (−7.68 − 13.3i)16-s + (−2.07 + 3.59i)17-s + (−1.55 − 2.69i)18-s − 7.26·19-s + ⋯
L(s)  = 1  + (0.969 + 1.67i)2-s − 0.787·3-s + (−1.37 + 2.38i)4-s + (0.165 − 0.287i)5-s + (−0.763 − 1.32i)6-s − 3.40·8-s − 0.379·9-s + 0.642·10-s − 0.411·11-s + (1.08 − 1.88i)12-s + (−0.0837 − 0.996i)13-s + (−0.130 + 0.226i)15-s + (−1.92 − 3.32i)16-s + (−0.503 + 0.871i)17-s + (−0.367 − 0.636i)18-s − 1.66·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0690 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0690 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0690 + 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0690 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406082 - 0.378940i\)
\(L(\frac12)\) \(\approx\) \(0.406082 - 0.378940i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.301 + 3.59i)T \)
good2 \( 1 + (-1.37 - 2.37i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 + (-0.370 + 0.641i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.36T + 11T^{2} \)
17 \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.26T + 19T^{2} \)
23 \( 1 + (-1.16 - 2.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 + 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.05 - 5.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.627 + 1.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.92 - 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.28 + 3.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.49 - 9.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.52T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + (-2.40 - 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.03 - 5.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.56 + 7.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (0.880 + 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.76 - 8.25i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42573656767079329621011058291, −10.48366998104307799288565533504, −9.022107548092504977592650881576, −8.336900303107059189435089451019, −7.54055159837041317109066366617, −6.38879519069210269946297319581, −5.93166218774981047064873079608, −5.10659574410804343930949318414, −4.32114817777628639813876778560, −2.99965305309110452730967924755, 0.22834720653696736611055329415, 2.04018719304843490286313911032, 2.92674361301654891588624847584, 4.33644863575091805584124804029, 4.91022792125433680349811350199, 6.00749219548933119545789280465, 6.68118517108898369927513505736, 8.657022768119527024547766376814, 9.408091596949703513863419327843, 10.56157846081933039527358792232

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