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Properties

Label 2-637-91.47-c1-0-22
Degree $2$
Conductor $637$
Sign $0.816 + 0.576i$
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  + (−1.84 + 0.493i)2-s + (−2.29 + 1.32i)3-s + (1.41 − 0.818i)4-s + (0.885 + 3.30i)5-s + (3.57 − 3.57i)6-s + (0.489 − 0.489i)8-s + (2.00 − 3.47i)9-s + (−3.26 − 5.65i)10-s + (1.66 + 0.445i)11-s + (−2.16 + 3.75i)12-s + (−3.57 + 0.501i)13-s + (−6.40 − 6.40i)15-s + (−2.29 + 3.97i)16-s + (−1.22 − 2.12i)17-s + (−1.97 + 7.38i)18-s + (−1.34 − 5.03i)19-s + ⋯
L(s)  = 1  + (−1.30 + 0.349i)2-s + (−1.32 + 0.764i)3-s + (0.708 − 0.409i)4-s + (0.396 + 1.47i)5-s + (1.45 − 1.45i)6-s + (0.173 − 0.173i)8-s + (0.668 − 1.15i)9-s + (−1.03 − 1.78i)10-s + (0.501 + 0.134i)11-s + (−0.625 + 1.08i)12-s + (−0.990 + 0.138i)13-s + (−1.65 − 1.65i)15-s + (−0.574 + 0.994i)16-s + (−0.297 − 0.515i)17-s + (−0.466 + 1.74i)18-s + (−0.309 − 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.816 + 0.576i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.816 + 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127051 - 0.0403352i\)
\(L(\frac12)\) \(\approx\) \(0.127051 - 0.0403352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.501i)T \)
good2 \( 1 + (1.84 - 0.493i)T + (1.73 - i)T^{2} \)
3 \( 1 + (2.29 - 1.32i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.885 - 3.30i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.66 - 0.445i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 + 5.03i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.97 + 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.184T + 29T^{2} \)
31 \( 1 + (2.46 + 0.659i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.0563 - 0.210i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.63 + 4.63i)T - 41iT^{2} \)
43 \( 1 - 0.562iT - 43T^{2} \)
47 \( 1 + (3.72 - 0.998i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.67 + 4.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.73 + 13.9i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.30 - 0.754i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.78 + 6.67i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.70 - 1.70i)T + 71iT^{2} \)
73 \( 1 + (3.15 - 11.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.504 + 0.504i)T - 83iT^{2} \)
89 \( 1 + (-7.20 + 1.92i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.0 - 12.0i)T - 97iT^{2} \)
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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

−10.32694313501137953117790219843, −9.805232930880811512100077274182, −9.130739087683442592194332599682, −7.72010649136222689074214508034, −6.71452471313321862621182681584, −6.50565595262209929116423269603, −5.15358266401002935646208237896, −4.04708377560845453280571279380, −2.34570601966030968368699394095, −0.15312492566944617006428528492, 1.12809305263329589377228772773, 1.89969537810872102082676779817, 4.42292989617219749463421907305, 5.40167772976532195300582907574, 6.13684325133852591421340575890, 7.37003942779547756465550866840, 8.148486541814068561100131216659, 9.011696501132850177535367135231, 9.806682945493876544280411235209, 10.53990343777070584113594189671

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