Properties

Label 2-637-49.37-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.681 - 0.732i$
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  + (0.445 − 0.304i)2-s + (0.759 − 0.704i)3-s + (−0.624 + 1.59i)4-s + (−1.39 − 0.430i)5-s + (0.124 − 0.545i)6-s + (−2.52 + 0.795i)7-s + (0.445 + 1.95i)8-s + (−0.143 + 1.92i)9-s + (−0.754 + 0.232i)10-s + (−0.221 − 2.95i)11-s + (0.646 + 1.64i)12-s + (−0.900 − 0.433i)13-s + (−0.883 + 1.12i)14-s + (−1.36 + 0.657i)15-s + (−1.71 − 1.58i)16-s + (−1.60 − 0.242i)17-s + ⋯
L(s)  = 1  + (0.315 − 0.214i)2-s + (0.438 − 0.406i)3-s + (−0.312 + 0.795i)4-s + (−0.624 − 0.192i)5-s + (0.0507 − 0.222i)6-s + (−0.953 + 0.300i)7-s + (0.157 + 0.689i)8-s + (−0.0479 + 0.640i)9-s + (−0.238 + 0.0735i)10-s + (−0.0667 − 0.890i)11-s + (0.186 + 0.475i)12-s + (−0.249 − 0.120i)13-s + (−0.236 + 0.299i)14-s + (−0.352 + 0.169i)15-s + (−0.428 − 0.397i)16-s + (−0.390 − 0.0588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.681 - 0.732i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.681 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240756 + 0.552940i\)
\(L(\frac12)\) \(\approx\) \(0.240756 + 0.552940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.52 - 0.795i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (-0.445 + 0.304i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-0.759 + 0.704i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (1.39 + 0.430i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.221 + 2.95i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (1.60 + 0.242i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (3.14 - 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.90 - 0.738i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (2.76 - 3.46i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-1.69 - 2.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.65 - 9.31i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.53 + 6.72i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.07 - 4.73i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-7.45 + 5.08i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (1.40 - 3.58i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-14.4 + 4.45i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (0.255 + 0.650i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (6.37 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.47 + 8.12i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-6.50 - 4.43i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (2.81 - 4.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.7 + 5.20i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.788 - 10.5i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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

−11.03369250967418897598862826870, −10.05634316468520638586749953007, −8.848474611743998095784166207910, −8.242268102614329913440805895737, −7.67725250619627683980556875474, −6.49012518163259398757224894622, −5.34760442981178188555966198473, −4.07859724118526021010132319922, −3.30317694165815289322668432492, −2.25108268600673800923114580110, 0.26469963943792578366925172917, 2.47565486457241097766078824559, 4.03033350163538592280154207507, 4.25823945597099731212713576485, 5.77885811404228676051142574398, 6.66255378505378605201731909168, 7.39060937659960106290518608187, 8.752370293296028382256584741182, 9.575585270284297305475579019688, 9.975135396141800034674605158543

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