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Properties

Label 2-207-69.68-c5-0-16
Degree $2$
Conductor $207$
Sign $0.0351 - 0.999i$
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  + 1.13i·2-s + 30.7·4-s − 11.6·5-s + 79.1i·7-s + 71.1i·8-s − 13.2i·10-s + 192.·11-s + 383.·13-s − 89.9·14-s + 901.·16-s − 1.91e3·17-s + 2.10e3i·19-s − 357.·20-s + 217. i·22-s + (2.01e3 + 1.53e3i)23-s + ⋯
L(s)  = 1  + 0.200i·2-s + 0.959·4-s − 0.208·5-s + 0.610i·7-s + 0.393i·8-s − 0.0418i·10-s + 0.478·11-s + 0.628·13-s − 0.122·14-s + 0.880·16-s − 1.60·17-s + 1.33i·19-s − 0.199·20-s + 0.0960i·22-s + (0.795 + 0.605i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0351 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0351 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.0351 - 0.999i$
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: \((2,\ 207,\ (\ :5/2),\ 0.0351 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.279539869\)
\(L(\frac12)\) \(\approx\) \(2.279539869\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.01e3 - 1.53e3i)T \)
good2 \( 1 - 1.13iT - 32T^{2} \)
5 \( 1 + 11.6T + 3.12e3T^{2} \)
7 \( 1 - 79.1iT - 1.68e4T^{2} \)
11 \( 1 - 192.T + 1.61e5T^{2} \)
13 \( 1 - 383.T + 3.71e5T^{2} \)
17 \( 1 + 1.91e3T + 1.41e6T^{2} \)
19 \( 1 - 2.10e3iT - 2.47e6T^{2} \)
29 \( 1 + 3.35e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.81e3T + 2.86e7T^{2} \)
37 \( 1 + 1.61e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.21e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.57e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.57e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.52e4T + 4.18e8T^{2} \)
59 \( 1 + 7.51e3iT - 7.14e8T^{2} \)
61 \( 1 - 5.16e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.86e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.42e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.87e4T + 2.07e9T^{2} \)
79 \( 1 + 7.16e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.92e4T + 3.93e9T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 - 4.45e4iT - 8.58e9T^{2} \)
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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

−11.60753779394186099775897497187, −11.06423446441491044601256185080, −9.804790121151526940319238481461, −8.632251973358609559240308843733, −7.71164899739375292976397111821, −6.49202489817122183388634319034, −5.82145702977190299852614905445, −4.18134167974018129794304552053, −2.74551204317889704236934051050, −1.48632123264263169083000483919, 0.67879649773964391006606130299, 2.13311097312687449189576170864, 3.47337986359439610826239615850, 4.72570182377160071898079303897, 6.47925696287084065946578515235, 6.91952178836310298981911321898, 8.241673705147775850768603074965, 9.344174160226020156003835948350, 10.68009578394447513321060122384, 11.13930656898421313054446877135

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