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Properties

Label 2-637-91.16-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.887 + 0.460i$
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  + 0.231·2-s + (1.66 − 2.87i)3-s − 1.94·4-s + (1.11 − 1.93i)5-s + (0.384 − 0.665i)6-s − 0.913·8-s + (−4.01 − 6.96i)9-s + (0.258 − 0.447i)10-s + (−1.66 + 2.87i)11-s + (−3.23 + 5.60i)12-s + (3.40 − 1.19i)13-s + (−3.70 − 6.41i)15-s + 3.68·16-s − 1.37·17-s + (−0.929 − 1.61i)18-s + (−1.61 − 2.80i)19-s + ⋯
L(s)  = 1  + 0.163·2-s + (0.959 − 1.66i)3-s − 0.973·4-s + (0.498 − 0.864i)5-s + (0.156 − 0.271i)6-s − 0.322·8-s + (−1.33 − 2.32i)9-s + (0.0816 − 0.141i)10-s + (−0.500 + 0.867i)11-s + (−0.933 + 1.61i)12-s + (0.943 − 0.330i)13-s + (−0.957 − 1.65i)15-s + 0.920·16-s − 0.333·17-s + (−0.219 − 0.379i)18-s + (−0.371 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403943 - 1.65696i\)
\(L(\frac12)\) \(\approx\) \(0.403943 - 1.65696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 + 1.19i)T \)
good2 \( 1 - 0.231T + 2T^{2} \)
3 \( 1 + (-1.66 + 2.87i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.11 + 1.93i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.838T + 23T^{2} \)
29 \( 1 + (-0.303 - 0.525i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.857 + 1.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.55T + 37T^{2} \)
41 \( 1 + (-4.58 - 7.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.615 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.814 + 1.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.19 + 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 + (2.73 + 4.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 + 8.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.60 + 4.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.98 - 3.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.64T + 83T^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + (-7.67 + 13.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.769304853354719030245042194154, −9.081596535025668380333527484875, −8.443546021555705839274921808846, −7.80557811362432173932639805831, −6.70777180246477469334581649491, −5.75185816800067071245697035260, −4.65177751704860485071780189775, −3.30003981581256588727328891705, −2.01390425101949095428953914717, −0.833518959541479747097546098540, 2.57254133532160647008559198775, 3.52730508566757892392740211944, 4.18333297005699709801180996678, 5.30104615489044544206338974141, 6.11769162829317079369545999624, 7.84673519925707827018415997840, 8.733246264473996378012981467003, 9.094998503416135366322575449691, 10.20680109589418905762695012921, 10.50968374037098922207711432643

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