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Properties

Label 2-639-71.20-c1-0-19
Degree $2$
Conductor $639$
Sign $0.999 + 0.0285i$
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  + (0.551 + 2.41i)2-s + (−3.73 + 1.79i)4-s − 2.28·5-s + (0.895 − 3.92i)7-s + (−3.30 − 4.14i)8-s + (−1.26 − 5.52i)10-s + (2.79 − 3.50i)11-s + (−1.66 − 2.09i)13-s + 9.97·14-s + (3.03 − 3.80i)16-s − 4.65·17-s + (−1.15 − 0.554i)19-s + (8.53 − 4.11i)20-s + (9.99 + 4.81i)22-s + (1.57 − 6.91i)23-s + ⋯
L(s)  = 1  + (0.389 + 1.70i)2-s + (−1.86 + 0.898i)4-s − 1.02·5-s + (0.338 − 1.48i)7-s + (−1.16 − 1.46i)8-s + (−0.399 − 1.74i)10-s + (0.841 − 1.05i)11-s + (−0.462 − 0.579i)13-s + 2.66·14-s + (0.759 − 0.951i)16-s − 1.12·17-s + (−0.264 − 0.127i)19-s + (1.90 − 0.919i)20-s + (2.13 + 1.02i)22-s + (0.329 − 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.999 + 0.0285i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.999 + 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908376 - 0.0129621i\)
\(L(\frac12)\) \(\approx\) \(0.908376 - 0.0129621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.368 + 8.41i)T \)
good2 \( 1 + (-0.551 - 2.41i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 + (-0.895 + 3.92i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-2.79 + 3.50i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.66 + 2.09i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 4.65T + 17T^{2} \)
19 \( 1 + (1.15 + 0.554i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.57 + 6.91i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.07 - 1.96i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-5.13 - 6.44i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (0.824 + 3.61i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-5.51 - 6.92i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (0.117 + 0.513i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (1.02 + 0.493i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (12.4 + 5.99i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-6.76 + 8.48i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (1.16 + 5.11i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-12.5 + 6.05i)T + (41.7 - 52.3i)T^{2} \)
73 \( 1 + (-0.609 - 2.67i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-2.05 - 2.57i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-1.33 + 1.66i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.32 + 2.56i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (7.35 - 9.22i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.73570546292566845354980387262, −9.293131448714517568893366454066, −8.254689948234348202478380735415, −7.933405905775688897907263817977, −6.83075957202120959701951831617, −6.50467181075764690835659551272, −4.99173284745645497860658173934, −4.29105670720394316669912753086, −3.51869397425794614229498383760, −0.44347813726705565725863702790, 1.76949288664073088122339288847, 2.60140937714872066707279092901, 3.99775160270348356361352623295, 4.50420857468346287733670409891, 5.65319455709425661469311696364, 7.09256670172613651500597477131, 8.305122833268856755356338408895, 9.327758607272048460721438103384, 9.593486015785536010239106482643, 11.03564164310951086143720106331

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