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Properties

Label 2-637-49.37-c1-0-23
Degree $2$
Conductor $637$
Sign $0.994 - 0.101i$
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  + (−0.610 + 0.416i)2-s + (−2.18 + 2.02i)3-s + (−0.531 + 1.35i)4-s + (−2.47 − 0.764i)5-s + (0.489 − 2.14i)6-s + (1.83 + 1.90i)7-s + (−0.567 − 2.48i)8-s + (0.439 − 5.86i)9-s + (1.83 − 0.564i)10-s + (−0.116 − 1.55i)11-s + (−1.58 − 4.03i)12-s + (−0.900 − 0.433i)13-s + (−1.91 − 0.399i)14-s + (6.96 − 3.35i)15-s + (−0.750 − 0.696i)16-s + (−5.91 − 0.892i)17-s + ⋯
L(s)  = 1  + (−0.431 + 0.294i)2-s + (−1.26 + 1.17i)3-s + (−0.265 + 0.676i)4-s + (−1.10 − 0.342i)5-s + (0.199 − 0.875i)6-s + (0.693 + 0.720i)7-s + (−0.200 − 0.879i)8-s + (0.146 − 1.95i)9-s + (0.579 − 0.178i)10-s + (−0.0351 − 0.469i)11-s + (−0.457 − 1.16i)12-s + (−0.249 − 0.120i)13-s + (−0.511 − 0.106i)14-s + (1.79 − 0.866i)15-s + (−0.187 − 0.174i)16-s + (−1.43 − 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.994 - 0.101i$
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: \((2,\ 637,\ (\ :1/2),\ 0.994 - 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284217 + 0.0144414i\)
\(L(\frac12)\) \(\approx\) \(0.284217 + 0.0144414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.83 - 1.90i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (0.610 - 0.416i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (2.18 - 2.02i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (2.47 + 0.764i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.116 + 1.55i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (5.91 + 0.892i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.151 - 0.262i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 0.470i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-2.28 + 2.86i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-3.43 - 5.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.53 - 3.90i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (0.928 + 4.06i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.421 + 1.84i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.766 - 0.522i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (3.77 - 9.61i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-2.99 + 0.925i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (3.37 + 8.60i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (6.59 + 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.25 + 2.82i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-8.48 - 5.78i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-1.75 + 3.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.4 - 6.45i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-1.22 + 16.3i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−10.79637017290868360926879180554, −9.626504019431692548173547631885, −8.769307786861768908106084427650, −8.246085209928017027543783460088, −7.07005952406028789666604587877, −6.01413899457229614517272332966, −4.69012164221285238663671991704, −4.49836752247923246274566214485, −3.20441125165342176221320386480, −0.29389964541160533993253525758, 0.912232594685121175851847521847, 2.14136572490024046820988770695, 4.33021789570435413525926074800, 5.03505256940072239906515132105, 6.26531365795708384314280339520, 7.06666877982096091786813362292, 7.73144165044480710844746976770, 8.662403877257688401489338566170, 10.04066376771965148082936616031, 10.93506278706659000034213421655

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