Properties

Label 2-207-23.22-c6-0-13
Degree $2$
Conductor $207$
Sign $0.327 - 0.944i$
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  − 6.36·2-s − 23.5·4-s + 60.2i·5-s − 233. i·7-s + 556.·8-s − 383. i·10-s + 1.39e3i·11-s − 1.28e3·13-s + 1.48e3i·14-s − 2.03e3·16-s + 2.82e3i·17-s − 1.01e4i·19-s − 1.41e3i·20-s − 8.89e3i·22-s + (3.98e3 − 1.14e4i)23-s + ⋯
L(s)  = 1  − 0.795·2-s − 0.367·4-s + 0.481i·5-s − 0.681i·7-s + 1.08·8-s − 0.383i·10-s + 1.04i·11-s − 0.584·13-s + 0.541i·14-s − 0.497·16-s + 0.574i·17-s − 1.48i·19-s − 0.176i·20-s − 0.835i·22-s + (0.327 − 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.327 - 0.944i$
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: \((2,\ 207,\ (\ :3),\ 0.327 - 0.944i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7869771410\)
\(L(\frac12)\) \(\approx\) \(0.7869771410\)
\(L(4)\) 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 \( 1 + (-3.98e3 + 1.14e4i)T \)
good2 \( 1 + 6.36T + 64T^{2} \)
5 \( 1 - 60.2iT - 1.56e4T^{2} \)
7 \( 1 + 233. iT - 1.17e5T^{2} \)
11 \( 1 - 1.39e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.28e3T + 4.82e6T^{2} \)
17 \( 1 - 2.82e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.01e4iT - 4.70e7T^{2} \)
29 \( 1 + 5.52e3T + 5.94e8T^{2} \)
31 \( 1 + 2.73e4T + 8.87e8T^{2} \)
37 \( 1 + 3.97e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.84e4T + 4.75e9T^{2} \)
43 \( 1 - 3.09e4iT - 6.32e9T^{2} \)
47 \( 1 + 6.27e4T + 1.07e10T^{2} \)
53 \( 1 + 5.61e4iT - 2.21e10T^{2} \)
59 \( 1 - 1.59e5T + 4.21e10T^{2} \)
61 \( 1 - 3.33e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.44e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.37e5T + 1.28e11T^{2} \)
73 \( 1 - 4.94e5T + 1.51e11T^{2} \)
79 \( 1 - 5.71e5iT - 2.43e11T^{2} \)
83 \( 1 - 7.20e5iT - 3.26e11T^{2} \)
89 \( 1 - 7.85e5iT - 4.96e11T^{2} \)
97 \( 1 - 2.74e5iT - 8.32e11T^{2} \)
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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

−11.08499039018389474123722103957, −10.42190741460800503991665862821, −9.564460275336435626899214533543, −8.659316744643802122781596761128, −7.40551157206230968914518633206, −6.88654230518902003321363154371, −5.01691592932043463231021941587, −4.08394323854680851111589208741, −2.36422868491571677314027484262, −0.848084292786411918805245062062, 0.39996455044474778203253768073, 1.66219513230084045664717093155, 3.39113636965012974464843425312, 4.88286954822731312352049135969, 5.79960574378701363348547816069, 7.41044917495950356998538659607, 8.365252036551311083470604954282, 9.076991010345102932723902597070, 9.880699722968982732505964373212, 10.98056962717578376650251282871

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