Properties

Label 2-637-49.4-c1-0-40
Degree $2$
Conductor $637$
Sign $0.756 + 0.653i$
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  + (1.12 + 0.767i)2-s + (−0.0507 − 0.0471i)3-s + (−0.0524 − 0.133i)4-s + (0.822 − 0.253i)5-s + (−0.0210 − 0.0920i)6-s + (−2.32 + 1.26i)7-s + (0.649 − 2.84i)8-s + (−0.223 − 2.98i)9-s + (1.12 + 0.345i)10-s + (0.329 − 4.40i)11-s + (−0.00363 + 0.00925i)12-s + (−0.900 + 0.433i)13-s + (−3.58 − 0.359i)14-s + (−0.0537 − 0.0258i)15-s + (2.70 − 2.51i)16-s + (7.62 − 1.14i)17-s + ⋯
L(s)  = 1  + (0.796 + 0.542i)2-s + (−0.0293 − 0.0272i)3-s + (−0.0262 − 0.0668i)4-s + (0.367 − 0.113i)5-s + (−0.00857 − 0.0375i)6-s + (−0.878 + 0.478i)7-s + (0.229 − 1.00i)8-s + (−0.0746 − 0.995i)9-s + (0.354 + 0.109i)10-s + (0.0994 − 1.32i)11-s + (−0.00104 + 0.00267i)12-s + (−0.249 + 0.120i)13-s + (−0.958 − 0.0961i)14-s + (−0.0138 − 0.00667i)15-s + (0.676 − 0.627i)16-s + (1.84 − 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.756 + 0.653i$
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: \((2,\ 637,\ (\ :1/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88943 - 0.703483i\)
\(L(\frac12)\) \(\approx\) \(1.88943 - 0.703483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.32 - 1.26i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
good2 \( 1 + (-1.12 - 0.767i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (0.0507 + 0.0471i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-0.822 + 0.253i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.329 + 4.40i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (-7.62 + 1.14i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.00360 + 0.00624i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.66 - 0.551i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (3.01 + 3.78i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (1.38 - 2.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.04 - 5.19i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-2.13 + 9.35i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.997 - 4.36i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-7.00 - 4.77i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (-0.126 - 0.321i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (2.64 + 0.816i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (-2.59 + 6.60i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (8.07 - 13.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.71 + 9.67i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (5.85 - 3.99i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-8.47 - 14.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.51 + 2.17i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.0121 + 0.162i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−10.29163280759817937136342085111, −9.516752360801296917152685769776, −8.980532249135521497524210539016, −7.57723216603821898978708013516, −6.55815688150904821603587115664, −5.80130257884190137431116962637, −5.40372625176899883127132085662, −3.78394357183939532349347963962, −3.12146452443859104627717894017, −0.904654687015108496038747794512, 1.92724486686308693235981084901, 3.03082291904189090390861132655, 4.03818192316051840822569968999, 5.02885273872231149665323264519, 5.86612999460171494822971076469, 7.30211110568127551961094536712, 7.77290469554851339461520022665, 9.155324009607595940380274631615, 10.12482450918693002129790189954, 10.56724342689875681603706152417

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