↑

Properties

Label 2-637-91.4-c1-0-11
Degree $2$
Conductor $637$
Sign $0.278 - 0.960i$
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  + 2.30i·2-s + (−0.736 − 1.27i)3-s − 3.30·4-s + (−0.733 + 0.423i)5-s + (2.93 − 1.69i)6-s − 3.00i·8-s + (0.414 − 0.718i)9-s + (−0.975 − 1.69i)10-s + (1.30 − 0.751i)11-s + (2.43 + 4.21i)12-s + (2.92 − 2.11i)13-s + (1.08 + 0.624i)15-s + 0.313·16-s + 2.07·17-s + (1.65 + 0.954i)18-s + (−0.0410 − 0.0237i)19-s + ⋯
L(s)  = 1  + 1.62i·2-s + (−0.425 − 0.736i)3-s − 1.65·4-s + (−0.328 + 0.189i)5-s + (1.19 − 0.692i)6-s − 1.06i·8-s + (0.138 − 0.239i)9-s + (−0.308 − 0.534i)10-s + (0.392 − 0.226i)11-s + (0.702 + 1.21i)12-s + (0.810 − 0.585i)13-s + (0.279 + 0.161i)15-s + 0.0782·16-s + 0.502·17-s + (0.389 + 0.225i)18-s + (−0.00942 − 0.00544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.962425 + 0.722943i\)
\(L(\frac12)\) \(\approx\) \(0.962425 + 0.722943i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 + 2.11i)T \)
good2 \( 1 - 2.30iT - 2T^{2} \)
3 \( 1 + (0.736 + 1.27i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.733 - 0.423i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.30 + 0.751i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + (0.0410 + 0.0237i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 + (0.679 - 1.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.80 - 3.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.70iT - 37T^{2} \)
41 \( 1 + (-8.67 - 5.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.311 + 0.180i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.35 - 2.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.64iT - 59T^{2} \)
61 \( 1 + (-2.26 + 3.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.3 + 7.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.85 + 3.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 + (0.369 - 0.213i)T + (48.5 - 84.0i)T^{2} \)
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

−10.88280072292119826145758560603, −9.477696570504865534383962501871, −8.729454374580036100239657726726, −7.71587589001812746101230336703, −7.23275839033477671014255157601, −6.28948996969251171522955139376, −5.80358049095509543459584835245, −4.63235155457724008056886878045, −3.33318677175599731727332108813, −0.994786452817822426793597153049, 1.06586463120164338046030277543, 2.50866659681620781884978587218, 3.86983036568855296025897571104, 4.34242314454134630905801923355, 5.39545333067639346326390721609, 6.77605180670282146905929917482, 8.147542259697561622161028094859, 9.142654224641971641421994704748, 9.792927317033908821577208204896, 10.57974345211293859523502005231

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