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Properties

Label 2-207-23.22-c6-0-10
Degree $2$
Conductor $207$
Sign $-0.638 + 0.769i$
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  − 10.8·2-s + 54.8·4-s + 162. i·5-s + 219. i·7-s + 100.·8-s − 1.77e3i·10-s + 1.60e3i·11-s − 3.92e3·13-s − 2.39e3i·14-s − 4.60e3·16-s + 6.86e3i·17-s + 1.47e3i·19-s + 8.90e3i·20-s − 1.75e4i·22-s + (−7.76e3 + 9.36e3i)23-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.856·4-s + 1.29i·5-s + 0.640i·7-s + 0.195·8-s − 1.77i·10-s + 1.20i·11-s − 1.78·13-s − 0.873i·14-s − 1.12·16-s + 1.39i·17-s + 0.215i·19-s + 1.11i·20-s − 1.64i·22-s + (−0.638 + 0.769i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.638 + 0.769i$
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: \((2,\ 207,\ (\ :3),\ -0.638 + 0.769i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.5429814765\)
\(L(\frac12)\) \(\approx\) \(0.5429814765\)
\(L(4)\) 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 \( 1 + (7.76e3 - 9.36e3i)T \)
good2 \( 1 + 10.8T + 64T^{2} \)
5 \( 1 - 162. iT - 1.56e4T^{2} \)
7 \( 1 - 219. iT - 1.17e5T^{2} \)
11 \( 1 - 1.60e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.92e3T + 4.82e6T^{2} \)
17 \( 1 - 6.86e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.47e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.85e4T + 5.94e8T^{2} \)
31 \( 1 - 2.70e4T + 8.87e8T^{2} \)
37 \( 1 - 7.43e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.33e4T + 4.75e9T^{2} \)
43 \( 1 - 6.86e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.05e5T + 1.07e10T^{2} \)
53 \( 1 - 2.57e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.58e5T + 4.21e10T^{2} \)
61 \( 1 - 2.24e5iT - 5.15e10T^{2} \)
67 \( 1 + 5.30e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.58e5T + 1.28e11T^{2} \)
73 \( 1 - 2.81e5T + 1.51e11T^{2} \)
79 \( 1 + 2.53e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.47e5iT - 3.26e11T^{2} \)
89 \( 1 + 2.60e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.39e6iT - 8.32e11T^{2} \)
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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

−11.70750874776757886919143068870, −10.48789342492696880621118933015, −10.01926014567549542223679560737, −9.187162566863868089541902473187, −7.79791756198715070814191655632, −7.30770330606957131869294134598, −6.15443984173564673258894141545, −4.49262232459747733046721028262, −2.67444733862285561625184732193, −1.76765281005619995160855978641, 0.36685325604497710694511007867, 0.69269479574991959575228246340, 2.34652299948498649968911825548, 4.33529111411665756525993044301, 5.33760144963015482587823200835, 7.07621301276313907407148588601, 7.87220945983549015003849719167, 8.847893600968181372959550587766, 9.484061679171193711005331480848, 10.38640839183109821027517269903

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