Properties

Label 2-639-213.53-c1-0-4
Degree $2$
Conductor $639$
Sign $-0.842 - 0.539i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
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  + (0.103 + 1.14i)2-s + (0.668 − 0.121i)4-s + (−0.774 + 1.06i)5-s + (−1.14 − 1.92i)7-s + (0.819 + 2.96i)8-s + (−1.29 − 0.776i)10-s + (−2.52 + 3.82i)11-s + (−4.09 + 2.70i)13-s + (2.08 − 1.51i)14-s + (−2.04 + 0.765i)16-s + (0.731 − 2.25i)17-s + (2.05 + 2.15i)19-s + (−0.388 + 0.806i)20-s + (−4.63 − 2.49i)22-s + (1.59 + 6.98i)23-s + ⋯
L(s)  = 1  + (0.0728 + 0.809i)2-s + (0.334 − 0.0606i)4-s + (−0.346 + 0.476i)5-s + (−0.433 − 0.726i)7-s + (0.289 + 1.04i)8-s + (−0.410 − 0.245i)10-s + (−0.761 + 1.15i)11-s + (−1.13 + 0.749i)13-s + (0.556 − 0.404i)14-s + (−0.510 + 0.191i)16-s + (0.177 − 0.546i)17-s + (0.471 + 0.493i)19-s + (−0.0868 + 0.180i)20-s + (−0.988 − 0.532i)22-s + (0.332 + 1.45i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.842 - 0.539i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.842 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350698 + 1.19842i\)
\(L(\frac12)\) \(\approx\) \(0.350698 + 1.19842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (-6.31 + 5.58i)T \)
good2 \( 1 + (-0.103 - 1.14i)T + (-1.96 + 0.357i)T^{2} \)
5 \( 1 + (0.774 - 1.06i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.14 + 1.92i)T + (-3.31 + 6.16i)T^{2} \)
11 \( 1 + (2.52 - 3.82i)T + (-4.32 - 10.1i)T^{2} \)
13 \( 1 + (4.09 - 2.70i)T + (5.10 - 11.9i)T^{2} \)
17 \( 1 + (-0.731 + 2.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.05 - 2.15i)T + (-0.852 + 18.9i)T^{2} \)
23 \( 1 + (-1.59 - 6.98i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-0.389 + 0.209i)T + (15.9 - 24.2i)T^{2} \)
31 \( 1 + (0.985 - 2.62i)T + (-23.3 - 20.3i)T^{2} \)
37 \( 1 + (-1.24 + 5.46i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (2.57 - 3.22i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-5.09 + 4.45i)T + (5.77 - 42.6i)T^{2} \)
47 \( 1 + (0.475 + 3.50i)T + (-45.3 + 12.5i)T^{2} \)
53 \( 1 + (1.22 + 0.221i)T + (49.6 + 18.6i)T^{2} \)
59 \( 1 + (0.190 + 4.24i)T + (-58.7 + 5.28i)T^{2} \)
61 \( 1 + (7.29 - 12.2i)T + (-28.9 - 53.7i)T^{2} \)
67 \( 1 + (2.16 + 11.9i)T + (-62.7 + 23.5i)T^{2} \)
73 \( 1 + (-10.1 + 0.916i)T + (71.8 - 13.0i)T^{2} \)
79 \( 1 + (-7.55 + 2.08i)T + (67.8 - 40.5i)T^{2} \)
83 \( 1 + (16.3 - 0.734i)T + (82.6 - 7.44i)T^{2} \)
89 \( 1 + (0.206 - 1.13i)T + (-83.3 - 31.2i)T^{2} \)
97 \( 1 + (-5.29 + 4.22i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.87668759340542487539344641083, −10.03630679261657521340910119156, −9.293016114706263521590896308884, −7.70719113219076925387437760249, −7.32698762462726400100923620916, −6.86412175919661822427627824960, −5.52220415822577562252807087894, −4.70586271644874875505688953304, −3.31851180411341076305741536528, −2.03170970556385842664724175968, 0.62590766200709061181354377787, 2.54647967732824588055203184060, 3.07918589752350525687160762984, 4.50019330448723417872756227690, 5.59703220056437606363397999441, 6.57142552832037604509952020823, 7.75938082996430921884556565891, 8.489323169953372611469424644060, 9.547491046636079970236027765997, 10.40311423050360423146570566620

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