Properties

Label 2-637-49.4-c1-0-24
Degree $2$
Conductor $637$
Sign $0.255 - 0.966i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 + 1.55i)2-s + (−1.64 − 1.52i)3-s + (2.04 + 5.21i)4-s + (3.15 − 0.973i)5-s + (−1.37 − 6.03i)6-s + (−0.157 + 2.64i)7-s + (−2.20 + 9.66i)8-s + (0.152 + 2.04i)9-s + (8.69 + 2.68i)10-s + (0.246 − 3.28i)11-s + (4.59 − 11.7i)12-s + (−0.900 + 0.433i)13-s + (−4.45 + 5.77i)14-s + (−6.68 − 3.21i)15-s + (−11.8 + 10.9i)16-s + (3.84 − 0.579i)17-s + ⋯
L(s)  = 1  + (1.61 + 1.09i)2-s + (−0.950 − 0.882i)3-s + (1.02 + 2.60i)4-s + (1.41 − 0.435i)5-s + (−0.562 − 2.46i)6-s + (−0.0595 + 0.998i)7-s + (−0.779 + 3.41i)8-s + (0.0509 + 0.680i)9-s + (2.75 + 0.848i)10-s + (0.0742 − 0.990i)11-s + (1.32 − 3.37i)12-s + (−0.249 + 0.120i)13-s + (−1.19 + 1.54i)14-s + (−1.72 − 0.831i)15-s + (−2.95 + 2.74i)16-s + (0.931 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59568 + 1.99956i\)
\(L(\frac12)\) \(\approx\) \(2.59568 + 1.99956i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.157 - 2.64i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
good2 \( 1 + (-2.27 - 1.55i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (1.64 + 1.52i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-3.15 + 0.973i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.246 + 3.28i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (-3.84 + 0.579i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-1.43 - 2.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.29 + 0.647i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-0.847 - 1.06i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-5.02 + 8.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 - 6.69i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-0.276 + 1.21i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (0.691 + 3.03i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.08 + 3.46i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (-1.88 - 4.80i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (6.62 + 2.04i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (-4.53 + 11.5i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-5.54 + 9.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.46 - 9.35i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.694 + 0.473i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (5.29 + 9.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.66 + 4.65i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.916 + 12.2i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + 3.19T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.46038888383561833922172182036, −9.860348928642591462913105185983, −8.616549314219444951090119636133, −7.78356309012914186772384591188, −6.58990290321791806390100458712, −5.89031157230270216647017983884, −5.76597590182843694796190680662, −4.87589849151981671584908633851, −3.22437968456606986469897900222, −1.95607182821944962808510074645, 1.45173220360302358790474647946, 2.73971499859071501904767839748, 3.96173622317529516973083403683, 4.81464914636533678114807791532, 5.46767798745277000154164373771, 6.27004387856293123840385637748, 7.10987497522876535744970976661, 9.677151679705321466390643659823, 10.03330969327763272734896659433, 10.42193940559808816487974307278

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