Properties

Label 2-639-213.92-c1-0-10
Degree $2$
Conductor $639$
Sign $0.950 - 0.312i$
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  + (1.46 − 0.968i)2-s + (0.428 − 1.00i)4-s + (0.365 + 0.503i)5-s + (1.05 + 2.80i)7-s + (0.285 + 1.57i)8-s + (1.02 + 0.384i)10-s + (0.685 + 5.06i)11-s + (−1.95 − 0.265i)13-s + (4.25 + 3.09i)14-s + (3.44 + 3.60i)16-s + (−0.337 − 1.03i)17-s + (−5.50 − 3.28i)19-s + (0.661 − 0.150i)20-s + (5.91 + 6.76i)22-s + (0.865 + 1.08i)23-s + ⋯
L(s)  = 1  + (1.03 − 0.684i)2-s + (0.214 − 0.501i)4-s + (0.163 + 0.225i)5-s + (0.397 + 1.05i)7-s + (0.100 + 0.556i)8-s + (0.323 + 0.121i)10-s + (0.206 + 1.52i)11-s + (−0.543 − 0.0736i)13-s + (1.13 + 0.827i)14-s + (0.862 + 0.902i)16-s + (−0.0817 − 0.251i)17-s + (−1.26 − 0.754i)19-s + (0.147 − 0.0337i)20-s + (1.26 + 1.44i)22-s + (0.180 + 0.226i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.950 - 0.312i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.950 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53896 + 0.406310i\)
\(L(\frac12)\) \(\approx\) \(2.53896 + 0.406310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.10 - 2.30i)T \)
good2 \( 1 + (-1.46 + 0.968i)T + (0.786 - 1.83i)T^{2} \)
5 \( 1 + (-0.365 - 0.503i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.05 - 2.80i)T + (-5.27 + 4.60i)T^{2} \)
11 \( 1 + (-0.685 - 5.06i)T + (-10.6 + 2.92i)T^{2} \)
13 \( 1 + (1.95 + 0.265i)T + (12.5 + 3.45i)T^{2} \)
17 \( 1 + (0.337 + 1.03i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.50 + 3.28i)T + (9.00 + 16.7i)T^{2} \)
23 \( 1 + (-0.865 - 1.08i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-4.52 + 5.17i)T + (-3.89 - 28.7i)T^{2} \)
31 \( 1 + (-1.48 - 1.42i)T + (1.39 + 30.9i)T^{2} \)
37 \( 1 + (-5.95 + 7.46i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-7.29 - 3.51i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.540 + 12.0i)T + (-42.8 - 3.85i)T^{2} \)
47 \( 1 + (10.8 - 0.977i)T + (46.2 - 8.39i)T^{2} \)
53 \( 1 + (-3.22 - 7.53i)T + (-36.6 + 38.3i)T^{2} \)
59 \( 1 + (-2.04 + 3.80i)T + (-32.5 - 49.2i)T^{2} \)
61 \( 1 + (0.550 - 1.46i)T + (-45.9 - 40.1i)T^{2} \)
67 \( 1 + (-0.630 - 0.269i)T + (46.3 + 48.4i)T^{2} \)
73 \( 1 + (-3.11 - 4.71i)T + (-28.6 + 67.1i)T^{2} \)
79 \( 1 + (-8.42 + 1.52i)T + (73.9 - 27.7i)T^{2} \)
83 \( 1 + (-2.00 - 1.07i)T + (45.7 + 69.2i)T^{2} \)
89 \( 1 + (8.45 - 3.61i)T + (61.5 - 64.3i)T^{2} \)
97 \( 1 + (6.69 + 13.9i)T + (-60.4 + 75.8i)T^{2} \)
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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

−10.85879099011080950683635053091, −9.937815859065836699430519059815, −8.974324911380047851205222897166, −8.036896526342146391545272639933, −6.88927662972636483395408390482, −5.82332584555448174432175175279, −4.78141017069502878582641195417, −4.26552461683605944348076655546, −2.57146700011246709839482525111, −2.19122050788409335467681006368, 1.10341595759804659825398700021, 3.19995904088734321601234924562, 4.20979705754584616436045173428, 4.95301563475291593407554491920, 6.05796463931721723932576793621, 6.65089692877164809241447145706, 7.74747165865480820416764237109, 8.538658305478408380862983325342, 9.749463210637241718841677272118, 10.63881514373826199342774590521

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