Properties

Label 2-207-69.68-c5-0-28
Degree $2$
Conductor $207$
Sign $-0.537 + 0.843i$
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  − 3.32i·2-s + 20.9·4-s − 58.3·5-s + 22.0i·7-s − 176. i·8-s + 194. i·10-s + 772.·11-s − 840.·13-s + 73.4·14-s + 84.6·16-s + 1.07e3·17-s + 726. i·19-s − 1.22e3·20-s − 2.56e3i·22-s + (−958. − 2.34e3i)23-s + ⋯
L(s)  = 1  − 0.587i·2-s + 0.654·4-s − 1.04·5-s + 0.170i·7-s − 0.972i·8-s + 0.614i·10-s + 1.92·11-s − 1.37·13-s + 0.100·14-s + 0.0826·16-s + 0.903·17-s + 0.461i·19-s − 0.683·20-s − 1.13i·22-s + (−0.377 − 0.925i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.537 + 0.843i$
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: \((2,\ 207,\ (\ :5/2),\ -0.537 + 0.843i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.730561407\)
\(L(\frac12)\) \(\approx\) \(1.730561407\)
\(L(\frac{7}{2})\) 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 \( 1 + (958. + 2.34e3i)T \)
good2 \( 1 + 3.32iT - 32T^{2} \)
5 \( 1 + 58.3T + 3.12e3T^{2} \)
7 \( 1 - 22.0iT - 1.68e4T^{2} \)
11 \( 1 - 772.T + 1.61e5T^{2} \)
13 \( 1 + 840.T + 3.71e5T^{2} \)
17 \( 1 - 1.07e3T + 1.41e6T^{2} \)
19 \( 1 - 726. iT - 2.47e6T^{2} \)
29 \( 1 + 8.09e3iT - 2.05e7T^{2} \)
31 \( 1 - 60.1T + 2.86e7T^{2} \)
37 \( 1 - 2.59e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.38e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.59e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.43e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.87e4T + 4.18e8T^{2} \)
59 \( 1 + 3.10e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.04e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.58e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.79e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.15e4T + 2.07e9T^{2} \)
79 \( 1 + 5.32e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.14e5T + 3.93e9T^{2} \)
89 \( 1 - 3.60e4T + 5.58e9T^{2} \)
97 \( 1 - 1.34e5iT - 8.58e9T^{2} \)
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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

−11.64324634582300056818357328362, −10.25870172862813114687372762965, −9.459906593055250039870408218055, −8.055293169997277629839564512653, −7.14397287408773825568707299204, −6.14418285398183217626556887396, −4.33231209888314896685521738534, −3.44614404973220139849139555986, −2.00988601073343043979745417499, −0.53838215571204589980027514782, 1.37436002238975208178271859499, 3.14585219775048144055507938859, 4.37576135053082132833998139030, 5.76668695658918607926945141983, 7.09350894764193068272046009128, 7.42169160638033881610296538050, 8.684079826731666004557738947081, 9.806911272343766121004168037276, 11.16848755110889520618642974052, 11.84706742775785471776725430266

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