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Properties

Label 2-637-49.37-c1-0-43
Degree $2$
Conductor $637$
Sign $0.416 + 0.908i$
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  + (0.168 − 0.114i)2-s + (2.17 − 2.01i)3-s + (−0.715 + 1.82i)4-s + (0.954 + 0.294i)5-s + (0.134 − 0.588i)6-s + (−0.383 − 2.61i)7-s + (0.179 + 0.786i)8-s + (0.432 − 5.77i)9-s + (0.194 − 0.0599i)10-s + (−0.109 − 1.45i)11-s + (2.12 + 5.40i)12-s + (−0.900 − 0.433i)13-s + (−0.364 − 0.396i)14-s + (2.66 − 1.28i)15-s + (−2.75 − 2.55i)16-s + (4.78 + 0.721i)17-s + ⋯
L(s)  = 1  + (0.119 − 0.0811i)2-s + (1.25 − 1.16i)3-s + (−0.357 + 0.911i)4-s + (0.427 + 0.131i)5-s + (0.0548 − 0.240i)6-s + (−0.145 − 0.989i)7-s + (0.0634 + 0.277i)8-s + (0.144 − 1.92i)9-s + (0.0615 − 0.0189i)10-s + (−0.0329 − 0.439i)11-s + (0.612 + 1.56i)12-s + (−0.249 − 0.120i)13-s + (−0.0975 − 0.105i)14-s + (0.689 − 0.331i)15-s + (−0.687 − 0.638i)16-s + (1.16 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.416 + 0.908i$
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: \((2,\ 637,\ (\ :1/2),\ 0.416 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94439 - 1.24720i\)
\(L(\frac12)\) \(\approx\) \(1.94439 - 1.24720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.383 + 2.61i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (-0.168 + 0.114i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-2.17 + 2.01i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-0.954 - 0.294i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.109 + 1.45i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (-4.78 - 0.721i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.372 + 0.645i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.24 + 0.639i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.92 + 2.41i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-3.10 - 5.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.27 - 5.78i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.39 - 6.11i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.925 - 4.05i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.59 - 1.08i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (1.35 - 3.44i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (4.50 - 1.39i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.97 + 5.02i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-6.31 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.88 + 8.62i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (12.5 + 8.58i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (1.38 - 2.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.0 - 6.29i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.106 - 1.41i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 3.09T + 97T^{2} \)
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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

−10.20506918589248318165254314842, −9.389739917832525532425903879045, −8.380141758036243285550646610819, −7.86343239628398125475884522777, −7.16299264555175886717989471067, −6.25889466225129070035826098597, −4.58246636485738020882358773287, −3.32231292441762437515525659537, −2.80492738628506726648627489200, −1.21225294652873303119678659587, 1.92874307922939694402390383442, 3.05164834892038636248622273776, 4.24506244577004722893540908715, 5.20002631447283716081886159149, 5.82928242442979208150842562240, 7.36297714514507852441320071132, 8.540571476804298477001558640870, 9.193069439317747698962171628408, 9.769079786024473474976926716240, 10.22007276652397425529083453539

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