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Properties

Label 2-207-69.68-c5-0-30
Degree $2$
Conductor $207$
Sign $-0.953 - 0.302i$
Analytic cond. $33.1994$
Root an. cond. $5.76189$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.4i·2-s − 77.1·4-s + 72.3·5-s + 9.31i·7-s + 471. i·8-s − 755. i·10-s + 308.·11-s + 245.·13-s + 97.3·14-s + 2.46e3·16-s − 762.·17-s − 2.84e3i·19-s − 5.58e3·20-s − 3.22e3i·22-s + (2.02e3 − 1.53e3i)23-s + ⋯
L(s)  = 1  − 1.84i·2-s − 2.41·4-s + 1.29·5-s + 0.0718i·7-s + 2.60i·8-s − 2.39i·10-s + 0.769·11-s + 0.403·13-s + 0.132·14-s + 2.40·16-s − 0.639·17-s − 1.81i·19-s − 3.12·20-s − 1.42i·22-s + (0.797 − 0.603i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.953 - 0.302i$
Analytic conductor: \(33.1994\)
Root analytic conductor: \(5.76189\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ -0.953 - 0.302i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.960890947\)
\(L(\frac12)\) \(\approx\) \(1.960890947\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.02e3 + 1.53e3i)T \)
good2 \( 1 + 10.4iT - 32T^{2} \)
5 \( 1 - 72.3T + 3.12e3T^{2} \)
7 \( 1 - 9.31iT - 1.68e4T^{2} \)
11 \( 1 - 308.T + 1.61e5T^{2} \)
13 \( 1 - 245.T + 3.71e5T^{2} \)
17 \( 1 + 762.T + 1.41e6T^{2} \)
19 \( 1 + 2.84e3iT - 2.47e6T^{2} \)
29 \( 1 + 6.02e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.57e3T + 2.86e7T^{2} \)
37 \( 1 - 9.44e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.27e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.41e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.76e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.43e4T + 4.18e8T^{2} \)
59 \( 1 + 5.42e3iT - 7.14e8T^{2} \)
61 \( 1 - 3.79e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.35e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.94e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.09e4T + 2.07e9T^{2} \)
79 \( 1 + 8.82e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.34e4T + 3.93e9T^{2} \)
89 \( 1 - 4.89e4T + 5.58e9T^{2} \)
97 \( 1 + 7.15e4iT - 8.58e9T^{2} \)
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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

−11.02651453508086960199992694342, −10.15088204719932501158636189937, −9.223337033107889517901168000863, −8.784532547273234639415650923352, −6.67425041720257511767883541706, −5.30118565930222413096887646710, −4.17518732958178740802216941249, −2.75127211504067553351099818886, −1.86821878700273341242917368626, −0.63155852647738929902189200712, 1.45099853720340358209621065307, 3.81179815613945127091655566379, 5.21856909953805617883505911255, 6.00219522640360810025589574915, 6.74365282465328246326214425029, 7.85993339595646487418829506236, 9.053984457682981218207776114864, 9.498732337338499728289780344840, 10.77969043025668856432396947882, 12.57606495286137805290928144504

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