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Properties

Label 2-207-207.22-c0-0-0
Degree $2$
Conductor $207$
Sign $-0.5 - 0.866i$
Analytic cond. $0.103306$
Root an. cond. $0.321413$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 + 1.62i)2-s + (−0.939 − 0.342i)3-s + (−1.26 + 2.19i)4-s + (−0.326 − 1.85i)6-s − 2.87·8-s + (0.766 + 0.642i)9-s + (1.93 − 1.62i)12-s + (0.939 − 1.62i)13-s + (−1.43 − 2.49i)16-s + (−0.326 + 1.85i)18-s + (−0.5 + 0.866i)23-s + (2.70 + 0.984i)24-s + (−0.5 − 0.866i)25-s + 3.53·26-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.939 + 1.62i)2-s + (−0.939 − 0.342i)3-s + (−1.26 + 2.19i)4-s + (−0.326 − 1.85i)6-s − 2.87·8-s + (0.766 + 0.642i)9-s + (1.93 − 1.62i)12-s + (0.939 − 1.62i)13-s + (−1.43 − 2.49i)16-s + (−0.326 + 1.85i)18-s + (−0.5 + 0.866i)23-s + (2.70 + 0.984i)24-s + (−0.5 − 0.866i)25-s + 3.53·26-s + (−0.500 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.103306\)
Root analytic conductor: \(0.321413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7918424775\)
\(L(\frac12)\) \(\approx\) \(0.7918424775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 0.347T + T^{2} \)
73 \( 1 - 0.347T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−13.16175875917504095248115236567, −12.40764686258829962326019994676, −11.36970335253433770080455449485, −10.01579630316506244830086625166, −8.314175605475365743692326558725, −7.69741351585888678091108067750, −6.50655848760131952298010046329, −5.80370916133098730207207173386, −4.95422883148007704837880648110, −3.59256274110444935456057849803, 1.73225099048033422953589967046, 3.67297728891292611189604572878, 4.47684233937124717347178729887, 5.61542004363749576648631745284, 6.63790554828572067453005974501, 8.964996214993168723873354205507, 9.821031458149153546557412828439, 10.79471411510718406762808592758, 11.43098815669188229370895671866, 12.06552224638023840982367365846

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