↑

Properties

Label 2-637-91.9-c1-0-18
Degree $2$
Conductor $637$
Sign $-0.898 - 0.438i$
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 + 2.01i)2-s + 2.30·3-s + (−1.71 + 2.97i)4-s + (−1.68 + 2.91i)5-s + (2.69 + 4.66i)6-s − 3.34·8-s + 2.33·9-s − 7.85·10-s + 2.33·11-s + (−3.96 + 6.87i)12-s + (0.408 − 3.58i)13-s + (−3.89 + 6.74i)15-s + (−0.466 − 0.808i)16-s + (2.72 − 4.72i)17-s + (2.71 + 4.70i)18-s − 7.16·19-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)2-s + 1.33·3-s + (−0.858 + 1.48i)4-s + (−0.753 + 1.30i)5-s + (1.09 + 1.90i)6-s − 1.18·8-s + 0.777·9-s − 2.48·10-s + 0.702·11-s + (−1.14 + 1.98i)12-s + (0.113 − 0.993i)13-s + (−1.00 + 1.74i)15-s + (−0.116 − 0.202i)16-s + (0.661 − 1.14i)17-s + (0.640 + 1.10i)18-s − 1.64·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.898 - 0.438i$
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: \((2,\ 637,\ (\ :1/2),\ -0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676485 + 2.92709i\)
\(L(\frac12)\) \(\approx\) \(0.676485 + 2.92709i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.408 + 3.58i)T \)
good2 \( 1 + (-1.16 - 2.01i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 + (1.68 - 2.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
17 \( 1 + (-2.72 + 4.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.16T + 19T^{2} \)
23 \( 1 + (-3.22 - 5.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.52 - 2.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.52 + 2.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.468 + 0.812i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 + 2.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.17 - 2.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.62 - 6.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.39T + 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.03 - 1.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.79 + 6.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 + (6.57 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.77 - 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.95939121275141527761186613479, −9.880552149897001796674441701050, −8.726902099445535695450253240832, −8.005633854625212988399601246203, −7.38035334435567519582866088482, −6.73093557353715359927386363843, −5.70098372095127071276472394142, −4.29659977058166801584180337669, −3.49634030185441026800058464471, −2.73676171278023224283348467809, 1.27037103182144382545279948801, 2.31397630469855369914719347403, 3.64912635468989981530883328805, 4.13669096871817833515954357938, 4.93278341097878210986699067128, 6.52837834644259138651705455720, 8.071469579731047115498061627822, 8.707407580995690726626819094821, 9.224696809189143224631623436320, 10.34646915106989752045656241382

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