Properties

Label 2-207-69.68-c5-0-1
Degree $2$
Conductor $207$
Sign $0.412 + 0.910i$
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  + 8.84i·2-s − 46.3·4-s + 32.1·5-s + 32.0i·7-s − 126. i·8-s + 284. i·10-s − 601.·11-s + 865.·13-s − 284.·14-s − 360.·16-s − 831.·17-s + 1.07e3i·19-s − 1.48e3·20-s − 5.32e3i·22-s + (−2.49e3 − 478. i)23-s + ⋯
L(s)  = 1  + 1.56i·2-s − 1.44·4-s + 0.574·5-s + 0.247i·7-s − 0.700i·8-s + 0.898i·10-s − 1.49·11-s + 1.42·13-s − 0.387·14-s − 0.352·16-s − 0.697·17-s + 0.681i·19-s − 0.831·20-s − 2.34i·22-s + (−0.982 − 0.188i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.412 + 0.910i$
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: \((2,\ 207,\ (\ :5/2),\ 0.412 + 0.910i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1935923046\)
\(L(\frac12)\) \(\approx\) \(0.1935923046\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.49e3 + 478. i)T \)
good2 \( 1 - 8.84iT - 32T^{2} \)
5 \( 1 - 32.1T + 3.12e3T^{2} \)
7 \( 1 - 32.0iT - 1.68e4T^{2} \)
11 \( 1 + 601.T + 1.61e5T^{2} \)
13 \( 1 - 865.T + 3.71e5T^{2} \)
17 \( 1 + 831.T + 1.41e6T^{2} \)
19 \( 1 - 1.07e3iT - 2.47e6T^{2} \)
29 \( 1 + 1.74e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.61e3T + 2.86e7T^{2} \)
37 \( 1 + 1.45e4iT - 6.93e7T^{2} \)
41 \( 1 - 4.60e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.61e4iT - 1.47e8T^{2} \)
47 \( 1 - 6.93e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.14e4T + 4.18e8T^{2} \)
59 \( 1 - 2.14e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.06e3iT - 8.44e8T^{2} \)
67 \( 1 - 6.90e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.05e3iT - 1.80e9T^{2} \)
73 \( 1 - 7.80e3T + 2.07e9T^{2} \)
79 \( 1 + 1.87e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.01e4T + 3.93e9T^{2} \)
89 \( 1 + 2.98e4T + 5.58e9T^{2} \)
97 \( 1 + 1.26e5iT - 8.58e9T^{2} \)
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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

−12.74719845206446088225793921821, −11.19298931272128141546955253778, −10.17475946110476803956976720805, −8.937731353177114832694832881842, −8.182318708567078786170126681855, −7.25197540739789705055802736283, −5.88136004211092207699820084068, −5.65729416677180410461433698106, −4.10449547694766657920088699969, −2.17141917237193371787863786167, 0.05513377685680449671320250498, 1.51806974121726572863045863582, 2.59864086415287464762494698924, 3.76967624278992819917220601210, 5.05572728142983964735651232463, 6.38895192273439894407740307725, 8.000755459049619139306022094680, 9.098920726701086111188667005121, 10.07271350240514520158866395594, 10.78707042250422007307803291337

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