Properties

Label 2-207-23.22-c6-0-47
Degree $2$
Conductor $207$
Sign $0.895 - 0.444i$
Analytic cond. $47.6211$
Root an. cond. $6.90081$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.4·2-s + 173.·4-s + 161. i·5-s − 585. i·7-s + 1.68e3·8-s + 2.48e3i·10-s + 2.12e3i·11-s + 1.92e3·13-s − 9.02e3i·14-s + 1.48e4·16-s − 1.37e3i·17-s + 3.65e3i·19-s + 2.79e4i·20-s + 3.26e4i·22-s + (1.09e4 − 5.40e3i)23-s + ⋯
L(s)  = 1  + 1.92·2-s + 2.70·4-s + 1.29i·5-s − 1.70i·7-s + 3.29·8-s + 2.48i·10-s + 1.59i·11-s + 0.877·13-s − 3.28i·14-s + 3.62·16-s − 0.279i·17-s + 0.532i·19-s + 3.49i·20-s + 3.06i·22-s + (0.895 − 0.444i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(47.6211\)
Root analytic conductor: \(6.90081\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :3),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(8.004416117\)
\(L(\frac12)\) \(\approx\) \(8.004416117\)
\(L(4)\) 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 \( 1 + (-1.09e4 + 5.40e3i)T \)
good2 \( 1 - 15.4T + 64T^{2} \)
5 \( 1 - 161. iT - 1.56e4T^{2} \)
7 \( 1 + 585. iT - 1.17e5T^{2} \)
11 \( 1 - 2.12e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.92e3T + 4.82e6T^{2} \)
17 \( 1 + 1.37e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.65e3iT - 4.70e7T^{2} \)
29 \( 1 + 6.03e3T + 5.94e8T^{2} \)
31 \( 1 - 3.25e3T + 8.87e8T^{2} \)
37 \( 1 + 2.55e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.01e5T + 4.75e9T^{2} \)
43 \( 1 - 8.34e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.19e5T + 1.07e10T^{2} \)
53 \( 1 - 8.70e4iT - 2.21e10T^{2} \)
59 \( 1 + 4.21e4T + 4.21e10T^{2} \)
61 \( 1 + 1.12e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.98e4iT - 9.04e10T^{2} \)
71 \( 1 + 2.34e5T + 1.28e11T^{2} \)
73 \( 1 + 7.52e4T + 1.51e11T^{2} \)
79 \( 1 + 5.39e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.21e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.34e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.22e6iT - 8.32e11T^{2} \)
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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

−11.46415937044973700755170607561, −10.67973243802045436061587731223, −10.16020530402347956104543413734, −7.48278995224925069930671730758, −7.07004029646504530710770106603, −6.26137632395996979213597152264, −4.72501491431918154766570448787, −3.91077372332492714875498419344, −2.98922380500229737507464298882, −1.60021611150044728161965996734, 1.29394646382771342217187984356, 2.68568423352350334486210935442, 3.74413374651567392146950308636, 5.19619722034779010396487099123, 5.53870007938834481877949040027, 6.50027014749811320887320457131, 8.304732135028462520440421865878, 8.970257800917671467676008848274, 10.91801761937211619269887466871, 11.71007443165486792954176779961

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