Properties

Label 2-207-69.68-c5-0-23
Degree $2$
Conductor $207$
Sign $-0.813 + 0.581i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6.00i·2-s − 4.04·4-s − 54.7·5-s + 232. i·7-s − 167. i·8-s + 328. i·10-s − 132.·11-s + 1.03e3·13-s + 1.39e3·14-s − 1.13e3·16-s − 771.·17-s − 2.41e3i·19-s + 221.·20-s + 794. i·22-s + (−14.4 − 2.53e3i)23-s + ⋯
L(s)  = 1  − 1.06i·2-s − 0.126·4-s − 0.979·5-s + 1.79i·7-s − 0.927i·8-s + 1.04i·10-s − 0.329·11-s + 1.69·13-s + 1.90·14-s − 1.11·16-s − 0.647·17-s − 1.53i·19-s + 0.123·20-s + 0.349i·22-s + (−0.00569 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.813 + 0.581i$
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: \((2,\ 207,\ (\ :5/2),\ -0.813 + 0.581i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.339977153\)
\(L(\frac12)\) \(\approx\) \(1.339977153\)
\(L(\frac{7}{2})\) 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 \( 1 + (14.4 + 2.53e3i)T \)
good2 \( 1 + 6.00iT - 32T^{2} \)
5 \( 1 + 54.7T + 3.12e3T^{2} \)
7 \( 1 - 232. iT - 1.68e4T^{2} \)
11 \( 1 + 132.T + 1.61e5T^{2} \)
13 \( 1 - 1.03e3T + 3.71e5T^{2} \)
17 \( 1 + 771.T + 1.41e6T^{2} \)
19 \( 1 + 2.41e3iT - 2.47e6T^{2} \)
29 \( 1 - 2.49e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.50e3T + 2.86e7T^{2} \)
37 \( 1 + 6.87e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.88e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.10e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.24e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.02e4T + 4.18e8T^{2} \)
59 \( 1 - 2.19e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.65e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.42e4iT - 1.35e9T^{2} \)
71 \( 1 + 93.9iT - 1.80e9T^{2} \)
73 \( 1 + 3.90e4T + 2.07e9T^{2} \)
79 \( 1 + 4.13e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.24e4T + 3.93e9T^{2} \)
89 \( 1 - 2.22e4T + 5.58e9T^{2} \)
97 \( 1 + 1.63e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.29835658533848069214865111309, −10.55205276893642818970147640563, −8.950248873947803409069604597357, −8.616966249420767045250339331174, −6.99414238511321934612193712078, −5.83279523439113797045558224978, −4.31428294960634316315943898027, −3.08415162599445325720225914308, −2.13642215866107986324781289173, −0.44116562964048617780494659616, 1.21804005325066940245042648172, 3.55305602634603758362636719868, 4.40356488795152408498906024846, 5.99579543096903135407563330966, 6.90375966977245281900406045100, 7.909819278622732497695407080897, 8.241390061459321714962451257614, 10.03543145808772072877141935992, 11.05418393715846842798565468413, 11.62230621409748299726173364178

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