↑

Properties

Label 2-637-91.89-c1-0-19
Degree $2$
Conductor $637$
Sign $-0.777 - 0.628i$
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  + (1.67 + 1.67i)2-s + (1.04 + 0.601i)3-s + 3.60i·4-s + (0.825 + 3.08i)5-s + (0.736 + 2.74i)6-s + (−2.68 + 2.68i)8-s + (−0.777 − 1.34i)9-s + (−3.77 + 6.54i)10-s + (−1.10 − 4.11i)11-s + (−2.16 + 3.75i)12-s + (3.48 + 0.920i)13-s + (−0.992 + 3.70i)15-s − 1.79·16-s − 1.44·17-s + (0.952 − 3.55i)18-s + (−2.42 − 0.649i)19-s + ⋯
L(s)  = 1  + (1.18 + 1.18i)2-s + (0.601 + 0.347i)3-s + 1.80i·4-s + (0.369 + 1.37i)5-s + (0.300 + 1.12i)6-s + (−0.950 + 0.950i)8-s + (−0.259 − 0.448i)9-s + (−1.19 + 2.06i)10-s + (−0.332 − 1.23i)11-s + (−0.625 + 1.08i)12-s + (0.966 + 0.255i)13-s + (−0.256 + 0.956i)15-s − 0.448·16-s − 0.350·17-s + (0.224 − 0.838i)18-s + (−0.556 − 0.149i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13354 + 3.20575i\)
\(L(\frac12)\) \(\approx\) \(1.13354 + 3.20575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.48 - 0.920i)T \)
good2 \( 1 + (-1.67 - 1.67i)T + 2iT^{2} \)
3 \( 1 + (-1.04 - 0.601i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.825 - 3.08i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.10 + 4.11i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 + (2.42 + 0.649i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.23iT - 23T^{2} \)
29 \( 1 + (-1.34 - 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.14 - 1.37i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.438 + 0.438i)T - 37iT^{2} \)
41 \( 1 + (5.04 + 1.35i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.46 + 3.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.39 - 1.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.79 - 6.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.43 - 1.43i)T + 59iT^{2} \)
61 \( 1 + (4.53 - 2.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.46 - 2.26i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.31 + 2.22i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.09 - 15.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.00 - 1.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.11 + 5.11i)T - 83iT^{2} \)
89 \( 1 + (4.95 + 4.95i)T + 89iT^{2} \)
97 \( 1 + (-1.62 - 6.06i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.90762155873998999478750997754, −10.17017302395552789493905483039, −8.701480806329887688734173626333, −8.299662812750412598604565697834, −6.93085107540990682914089266429, −6.38784300828902565507073484586, −5.78105019567286752367410025140, −4.38324528439306708866916825579, −3.38527485909778066848201171643, −2.80683934906923212799248745898, 1.46356168347712078315503424466, 2.19886902641182525536732185422, 3.48754203456050373539790278626, 4.66856476623584455174871816698, 5.14221267529886539869554370342, 6.25905407321179438186127109982, 7.83109796115194892503155900402, 8.578148630712753109303378955344, 9.594665504351626240022524776687, 10.34343502469158653719182845908

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