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Properties

Label 2-637-91.74-c1-0-12
Degree $2$
Conductor $637$
Sign $0.894 + 0.446i$
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  − 1.90·2-s + (0.214 + 0.371i)3-s + 1.63·4-s + (−0.736 − 1.27i)5-s + (−0.408 − 0.707i)6-s + 0.702·8-s + (1.40 − 2.43i)9-s + (1.40 + 2.43i)10-s + (2.19 + 3.80i)11-s + (0.349 + 0.605i)12-s + (−2.69 + 2.39i)13-s + (0.315 − 0.546i)15-s − 4.60·16-s + 1.20·17-s + (−2.68 + 4.64i)18-s + (1.62 − 2.80i)19-s + ⋯
L(s)  = 1  − 1.34·2-s + (0.123 + 0.214i)3-s + 0.815·4-s + (−0.329 − 0.570i)5-s + (−0.166 − 0.288i)6-s + 0.248·8-s + (0.469 − 0.813i)9-s + (0.443 + 0.768i)10-s + (0.662 + 1.14i)11-s + (0.100 + 0.174i)12-s + (−0.748 + 0.663i)13-s + (0.0814 − 0.141i)15-s − 1.15·16-s + 0.291·17-s + (−0.632 + 1.09i)18-s + (0.371 − 0.644i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705747 - 0.166416i\)
\(L(\frac12)\) \(\approx\) \(0.705747 - 0.166416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.69 - 2.39i)T \)
good2 \( 1 + 1.90T + 2T^{2} \)
3 \( 1 + (-0.214 - 0.371i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.736 + 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 + (-1.62 + 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.43T + 23T^{2} \)
29 \( 1 + (0.0837 - 0.145i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.62 + 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
41 \( 1 + (-2.58 + 4.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0113 + 0.0197i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + (-5.77 + 9.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.98 - 8.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + 6.55T + 89T^{2} \)
97 \( 1 + (-1.74 - 3.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.01825089649615216348260010088, −9.543687523807550204171028763281, −9.074059497492519170512900797581, −7.995822395142424096115357289495, −7.26454040672391785057778040010, −6.43297120665741401742128476786, −4.68533012903465507393063992099, −4.11072420633955706857468804404, −2.18762863127060956220944978898, −0.811997872974530704723468565031, 1.06037455747608530093793971085, 2.52479101997313739219425835019, 3.86071348566735309624975574246, 5.28438768790229430177068411622, 6.55954873074569235649931196735, 7.51138206892934270889990625400, 7.993593341207264705586625776028, 8.807797872724497893207643247131, 9.890260707294501151215251783050, 10.39224013085585195437956401213

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