↑

Properties

Label 2-637-91.31-c1-0-15
Degree $2$
Conductor $637$
Sign $0.966 - 0.256i$
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  + (−1.16 − 0.312i)2-s + (1.96 + 1.13i)3-s + (−0.466 − 0.269i)4-s + (0.224 − 0.837i)5-s + (−1.94 − 1.94i)6-s + (2.17 + 2.17i)8-s + (1.08 + 1.87i)9-s + (−0.523 + 0.907i)10-s + (2.53 − 0.678i)11-s + (−0.612 − 1.06i)12-s + (0.104 + 3.60i)13-s + (1.39 − 1.39i)15-s + (−1.31 − 2.27i)16-s + (−1.52 + 2.63i)17-s + (−0.678 − 2.53i)18-s + (−0.0381 + 0.142i)19-s + ⋯
L(s)  = 1  + (−0.825 − 0.221i)2-s + (1.13 + 0.656i)3-s + (−0.233 − 0.134i)4-s + (0.100 − 0.374i)5-s + (−0.793 − 0.793i)6-s + (0.767 + 0.767i)8-s + (0.361 + 0.626i)9-s + (−0.165 + 0.286i)10-s + (0.763 − 0.204i)11-s + (−0.176 − 0.306i)12-s + (0.0289 + 0.999i)13-s + (0.359 − 0.359i)15-s + (−0.328 − 0.569i)16-s + (−0.369 + 0.640i)17-s + (−0.160 − 0.597i)18-s + (−0.00875 + 0.0326i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35923 + 0.177390i\)
\(L(\frac12)\) \(\approx\) \(1.35923 + 0.177390i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.104 - 3.60i)T \)
good2 \( 1 + (1.16 + 0.312i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-1.96 - 1.13i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.224 + 0.837i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.53 + 0.678i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.52 - 2.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0381 - 0.142i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.63 + 3.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + (-9.25 + 2.47i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.741 - 2.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.27 - 2.27i)T + 41iT^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (-7.12 - 1.90i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.71 - 2.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.34 - 12.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.97 - 4.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.384 + 1.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.10 - 4.10i)T - 71iT^{2} \)
73 \( 1 + (2.53 + 9.46i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.79 + 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.5 - 10.5i)T + 83iT^{2} \)
89 \( 1 + (-4.63 - 1.24i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.44 + 4.44i)T + 97iT^{2} \)
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

−10.32187756173859857422445288673, −9.471630453137274911862734321583, −8.816534790088663700761520540873, −8.671632247446504221876244044019, −7.44763007880854638071420400987, −6.19995224262935331913928654784, −4.70707761679453761507960047083, −4.08564674458373390215608072836, −2.68164146784126454294537312258, −1.31818391613861815618712459759, 1.10750557085766123359938114306, 2.63181604020069572992701114946, 3.60757638053355440205518680207, 4.96936092523411778185394810580, 6.59146777701464962221071867757, 7.29585205521662826098651522789, 7.977859295222719261318861926215, 8.807829940023249522704849744746, 9.323060425284445339105890478258, 10.24858827847097053556906585715

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