Properties

Label 2-637-49.44-c1-0-49
Degree $2$
Conductor $637$
Sign $-0.352 + 0.935i$
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  + (0.893 − 0.134i)2-s + (1.14 + 0.782i)3-s + (−1.13 + 0.349i)4-s + (0.316 − 4.22i)5-s + (1.13 + 0.544i)6-s + (−2.63 + 0.243i)7-s + (−2.59 + 1.24i)8-s + (−0.390 − 0.994i)9-s + (−0.285 − 3.81i)10-s + (0.753 − 1.91i)11-s + (−1.57 − 0.485i)12-s + (−0.623 − 0.781i)13-s + (−2.31 + 0.572i)14-s + (3.66 − 4.60i)15-s + (−0.188 + 0.128i)16-s + (−3.82 + 3.55i)17-s + ⋯
L(s)  = 1  + (0.631 − 0.0951i)2-s + (0.662 + 0.451i)3-s + (−0.565 + 0.174i)4-s + (0.141 − 1.88i)5-s + (0.461 + 0.222i)6-s + (−0.995 + 0.0920i)7-s + (−0.916 + 0.441i)8-s + (−0.130 − 0.331i)9-s + (−0.0904 − 1.20i)10-s + (0.227 − 0.578i)11-s + (−0.454 − 0.140i)12-s + (−0.172 − 0.216i)13-s + (−0.620 + 0.152i)14-s + (0.947 − 1.18i)15-s + (−0.0472 + 0.0322i)16-s + (−0.928 + 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.352 + 0.935i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (534, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824817 - 1.19252i\)
\(L(\frac12)\) \(\approx\) \(0.824817 - 1.19252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 - 0.243i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-0.893 + 0.134i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (-1.14 - 0.782i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (-0.316 + 4.22i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-0.753 + 1.91i)T + (-8.06 - 7.48i)T^{2} \)
17 \( 1 + (3.82 - 3.55i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-3.24 + 5.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.00756 - 0.00701i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (1.28 + 5.64i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.56 - 7.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.37 - 0.731i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.45 + 1.18i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-2.44 - 1.17i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.39 - 0.210i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (6.50 - 2.00i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.315 + 4.21i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-11.5 - 3.57i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-5.71 - 9.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.48 + 10.9i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.16 - 0.326i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-6.68 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.35 - 10.4i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (0.898 + 2.29i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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

−9.832496868492085606741227236844, −9.287836940725297863459520153759, −8.752181869153619000684072234829, −8.196149733580337680902660621668, −6.39698912606542947318510827709, −5.49230452188530297640231435042, −4.54949785898378291792440086147, −3.84830541057513357631324337330, −2.78758436838947790099660059911, −0.58709272281626020565590139644, 2.35391990262944743518454862252, 3.15026932168156192375556362214, 4.03447048640776764325129198310, 5.53024967959369393625837770318, 6.52597883975671056187118372718, 7.06928991079337286490578153054, 8.006003316965823902008886454757, 9.451087162275838795608344428810, 9.768670607879758433573180298344, 10.79640626219327491988345772182

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