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Properties

Label 2-207-69.68-c5-0-20
Degree $2$
Conductor $207$
Sign $-0.819 - 0.572i$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.819 - 0.572i$
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: \((2,\ 207,\ (\ :5/2),\ -0.819 - 0.572i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.934092330\)
\(L(\frac12)\) \(\approx\) \(2.934092330\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.63e5iT - 8.58e9T^{2} \)
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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

−11.79637564786021510512544126319, −11.12575598607036523204561028002, −9.495277295468206053040705320012, −8.874668951625613679942653642990, −7.928984311031764966038293553488, −6.33726551568938969183963123953, −6.01124057916614932612463902210, −5.06497079533193564472704362845, −2.89897895437190919162779277546, −1.69816623716090162707092711935, 0.955832352805133440272157096007, 1.62813592498769156025991469345, 3.37072333818579177116746127027, 4.14956033355480128373254870942, 6.01406396186878531290268404248, 6.87269608938495698348858446595, 8.226545935688385146969805510811, 9.662976646598355236763364228900, 10.40791500554484913399955546635, 10.80638289054084965805996287189

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