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Properties

Label 2-207-23.18-c1-0-2
Degree $2$
Conductor $207$
Sign $0.933 + 0.357i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
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  + (−1.54 − 0.454i)2-s + (0.506 + 0.325i)4-s + (−0.0749 + 0.521i)5-s + (0.200 + 0.439i)7-s + (1.47 + 1.70i)8-s + (0.353 − 0.772i)10-s + (3.21 − 0.943i)11-s + (0.853 − 1.86i)13-s + (−0.111 − 0.772i)14-s + (−2.01 − 4.40i)16-s + (0.919 − 0.590i)17-s + (6.19 + 3.98i)19-s + (−0.207 + 0.239i)20-s − 5.40·22-s + (1.81 − 4.43i)23-s + ⋯
L(s)  = 1  + (−1.09 − 0.321i)2-s + (0.253 + 0.162i)4-s + (−0.0335 + 0.233i)5-s + (0.0759 + 0.166i)7-s + (0.522 + 0.602i)8-s + (0.111 − 0.244i)10-s + (0.968 − 0.284i)11-s + (0.236 − 0.518i)13-s + (−0.0296 − 0.206i)14-s + (−0.502 − 1.10i)16-s + (0.223 − 0.143i)17-s + (1.42 + 0.913i)19-s + (−0.0464 + 0.0536i)20-s − 1.15·22-s + (0.378 − 0.925i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ 0.933 + 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705678 - 0.130388i\)
\(L(\frac12)\) \(\approx\) \(0.705678 - 0.130388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + (-1.81 + 4.43i)T \)
good2 \( 1 + (1.54 + 0.454i)T + (1.68 + 1.08i)T^{2} \)
5 \( 1 + (0.0749 - 0.521i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.200 - 0.439i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-3.21 + 0.943i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.853 + 1.86i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.919 + 0.590i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-6.19 - 3.98i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-3.03 + 1.94i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (3.65 + 4.21i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.491 - 3.42i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.47 - 10.2i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (5.05 - 5.83i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (-3.06 - 6.71i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-0.390 + 0.854i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (3.75 + 4.33i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (12.3 + 3.62i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-5.99 - 1.76i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.00713 + 0.00458i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-4.29 + 9.40i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (0.861 + 5.99i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-1.84 + 2.13i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (2.61 - 18.1i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.94648101600379368890586934003, −11.24199164368682478942425288529, −10.21273641935568310475720811000, −9.459239870213655236542824360213, −8.495228880232599600425083412968, −7.63152192188936622851485205192, −6.28325714472482894145501891983, −4.89426480646686061411467322559, −3.14893437940180377521576531756, −1.25290623750613008921047801779, 1.27484762866727987925626997296, 3.67257092463216130027519618486, 5.05818457285178730007853360734, 6.77576943705028498118625827410, 7.39919569789566400256013416684, 8.758357926521607511170514598657, 9.209819119744182724576872403296, 10.23659256201528682951139316587, 11.33503646479017205156381833104, 12.32660083574496253063425858473

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