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Properties

Label 2-637-91.25-c1-0-20
Degree $2$
Conductor $637$
Sign $0.998 - 0.0623i$
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  + (−1.73 + i)2-s + (1 − 1.73i)3-s + (0.999 − 1.73i)4-s + (−0.866 + 0.5i)5-s + 3.99i·6-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (1.73 + i)11-s + (−2 − 3.46i)12-s + (−2 + 3i)13-s + 1.99i·15-s + (1.99 + 3.46i)16-s + (3 − 5.19i)17-s + (1.73 + 0.999i)18-s + (2.59 − 1.5i)19-s + 1.99i·20-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.577 − 0.999i)3-s + (0.499 − 0.866i)4-s + (−0.387 + 0.223i)5-s + 1.63i·6-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (0.522 + 0.301i)11-s + (−0.577 − 0.999i)12-s + (−0.554 + 0.832i)13-s + 0.516i·15-s + (0.499 + 0.866i)16-s + (0.727 − 1.26i)17-s + (0.408 + 0.235i)18-s + (0.596 − 0.344i)19-s + 0.447i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.998 - 0.0623i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.998 - 0.0623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917734 + 0.0286180i\)
\(L(\frac12)\) \(\approx\) \(0.917734 + 0.0286180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3i)T \)
good2 \( 1 + (1.73 - i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.866 - 0.5i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 + 1.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.19 + 3i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-9.52 + 5.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 + (-7.79 - 4.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + (4.33 - 2.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9iT - 97T^{2} \)
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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

−10.20764693425916979298154484828, −9.350920872422043463201957766150, −8.832142264821832392307059781501, −7.69886359132345263770259541458, −7.24512420702448550969365528245, −6.87379725261714268445240803608, −5.42423301420494239919116561392, −3.84000591229597774409823339438, −2.35599495893713423354033459174, −0.976470815622892274161568489520, 1.04894841684322899871543530561, 2.74411017026177378895868327267, 3.65806054521207835139998591699, 4.75859115547961153425264867321, 6.05989271097388934422250962513, 7.66472001774795450148252438123, 8.276307414051792271246565881010, 8.983929542699496932587722019036, 9.821075706503844788815199790553, 10.26747926127694290272851262534

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