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Properties

Label 2-207-23.22-c6-0-5
Degree $2$
Conductor $207$
Sign $-0.992 + 0.126i$
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  + 8.53·2-s + 8.90·4-s + 233. i·5-s − 155. i·7-s − 470.·8-s + 1.99e3i·10-s + 880. i·11-s + 3.10e3·13-s − 1.33e3i·14-s − 4.58e3·16-s + 5.58e3i·17-s − 9.60e3i·19-s + 2.07e3i·20-s + 7.51e3i·22-s + (−1.20e4 + 1.53e3i)23-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.139·4-s + 1.86i·5-s − 0.454i·7-s − 0.918·8-s + 1.99i·10-s + 0.661i·11-s + 1.41·13-s − 0.485i·14-s − 1.11·16-s + 1.13i·17-s − 1.40i·19-s + 0.259i·20-s + 0.705i·22-s + (−0.992 + 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.992 + 0.126i$
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: \((2,\ 207,\ (\ :3),\ -0.992 + 0.126i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.247627085\)
\(L(\frac12)\) \(\approx\) \(1.247627085\)
\(L(4)\) 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 \( 1 + (1.20e4 - 1.53e3i)T \)
good2 \( 1 - 8.53T + 64T^{2} \)
5 \( 1 - 233. iT - 1.56e4T^{2} \)
7 \( 1 + 155. iT - 1.17e5T^{2} \)
11 \( 1 - 880. iT - 1.77e6T^{2} \)
13 \( 1 - 3.10e3T + 4.82e6T^{2} \)
17 \( 1 - 5.58e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.60e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.49e4T + 5.94e8T^{2} \)
31 \( 1 + 4.97e4T + 8.87e8T^{2} \)
37 \( 1 - 7.10e3iT - 2.56e9T^{2} \)
41 \( 1 + 3.96e4T + 4.75e9T^{2} \)
43 \( 1 + 9.71e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.44e4T + 1.07e10T^{2} \)
53 \( 1 + 1.96e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.77e5T + 4.21e10T^{2} \)
61 \( 1 - 3.43e5iT - 5.15e10T^{2} \)
67 \( 1 - 3.00e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.04e5T + 1.28e11T^{2} \)
73 \( 1 + 4.66e5T + 1.51e11T^{2} \)
79 \( 1 + 7.80e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.08e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.46e5iT - 4.96e11T^{2} \)
97 \( 1 - 6.91e5iT - 8.32e11T^{2} \)
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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

−11.78200370511893164472204254695, −10.94930147671933307804573694931, −10.23142190025211828784303684981, −8.881724448674826017416752226137, −7.38496668813774874542431800704, −6.53897014511706692885118059147, −5.69069113293222826239577421148, −4.01738864822758843282245601241, −3.46865481838963424403507359528, −2.13025719800683219570127489548, 0.22325942717665385974244524599, 1.58239684888138495002793781642, 3.49484818397365737319475597680, 4.39039766768493000374903883557, 5.54547448370047766166070707385, 5.92922464568925042957551436268, 8.024558420431154063506974375424, 8.839347878998534260575335413915, 9.492680248933965854555896288689, 11.25778479214670448434143420363

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