↑

Properties

Label 2-637-13.9-c1-0-31
Degree $2$
Conductor $637$
Sign $-0.0662 + 0.997i$
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  + (0.952 − 1.65i)2-s + (0.214 − 0.371i)3-s + (−0.815 − 1.41i)4-s + 1.47·5-s + (−0.408 − 0.707i)6-s + 0.702·8-s + (1.40 + 2.43i)9-s + (1.40 − 2.43i)10-s + (2.19 − 3.80i)11-s − 0.698·12-s + (−2.69 + 2.39i)13-s + (0.315 − 0.546i)15-s + (2.30 − 3.98i)16-s + (−0.601 − 1.04i)17-s + 5.36·18-s + (1.62 + 2.80i)19-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)2-s + (0.123 − 0.214i)3-s + (−0.407 − 0.706i)4-s + 0.658·5-s + (−0.166 − 0.288i)6-s + 0.248·8-s + (0.469 + 0.813i)9-s + (0.443 − 0.768i)10-s + (0.662 − 1.14i)11-s − 0.201·12-s + (−0.748 + 0.663i)13-s + (0.0814 − 0.141i)15-s + (0.575 − 0.996i)16-s + (−0.145 − 0.252i)17-s + 1.26·18-s + (0.371 + 0.644i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0662 + 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0662 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79863 - 1.92200i\)
\(L(\frac12)\) \(\approx\) \(1.79863 - 1.92200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.69 - 2.39i)T \)
good2 \( 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.214 + 0.371i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.47T + 5T^{2} \)
11 \( 1 + (-2.19 + 3.80i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.62 - 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0837 - 0.145i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.24T + 31T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.58 + 4.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0113 + 0.0197i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 0.141T + 53T^{2} \)
59 \( 1 + (2.67 + 4.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.77 - 9.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.98 - 8.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 0.774T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (-3.27 + 5.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.74 - 3.02i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.47243284934373811217494805799, −9.822541607415890265045814125430, −8.831231251741952447931153830158, −7.70008068916775842636549925169, −6.70284285982844094072486676390, −5.50348382428050391925076611332, −4.60471264221294279352924935556, −3.52613901911939569504501647789, −2.39125458713730210177750496910, −1.46713675594188325938413675365, 1.76103362439323387243986848632, 3.52435549322508809895105736344, 4.59413447888692580824827373955, 5.36913233024785055545320841917, 6.36669834854645623675837926193, 7.08187092447380363169152024318, 7.77785038667649480175568226558, 9.233306508850426564696787788351, 9.668520040290791458923444913427, 10.62497398648654110588637193324

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