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Properties

Label 2-207-69.68-c5-0-13
Degree $2$
Conductor $207$
Sign $-0.273 - 0.961i$
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  + 10.4i·2-s − 77.3·4-s − 44.1·5-s − 173. i·7-s − 474. i·8-s − 461. i·10-s − 517.·11-s − 1.13e3·13-s + 1.81e3·14-s + 2.48e3·16-s + 1.73e3·17-s + 2.75e3i·19-s + 3.41e3·20-s − 5.41e3i·22-s + (2.39e3 + 843. i)23-s + ⋯
L(s)  = 1  + 1.84i·2-s − 2.41·4-s − 0.789·5-s − 1.33i·7-s − 2.62i·8-s − 1.45i·10-s − 1.29·11-s − 1.85·13-s + 2.47·14-s + 2.43·16-s + 1.45·17-s + 1.74i·19-s + 1.90·20-s − 2.38i·22-s + (0.943 + 0.332i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.273 - 0.961i$
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: \((2,\ 207,\ (\ :5/2),\ -0.273 - 0.961i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8276373963\)
\(L(\frac12)\) \(\approx\) \(0.8276373963\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.39e3 - 843. i)T \)
good2 \( 1 - 10.4iT - 32T^{2} \)
5 \( 1 + 44.1T + 3.12e3T^{2} \)
7 \( 1 + 173. iT - 1.68e4T^{2} \)
11 \( 1 + 517.T + 1.61e5T^{2} \)
13 \( 1 + 1.13e3T + 3.71e5T^{2} \)
17 \( 1 - 1.73e3T + 1.41e6T^{2} \)
19 \( 1 - 2.75e3iT - 2.47e6T^{2} \)
29 \( 1 - 3.33e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.81e3T + 2.86e7T^{2} \)
37 \( 1 + 5.31e3iT - 6.93e7T^{2} \)
41 \( 1 + 8.22e3iT - 1.15e8T^{2} \)
43 \( 1 - 5.22e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.03e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.97e4T + 4.18e8T^{2} \)
59 \( 1 + 4.80e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.11e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.00e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.11e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.10e3T + 2.07e9T^{2} \)
79 \( 1 + 7.86e4iT - 3.07e9T^{2} \)
83 \( 1 + 3.78e4T + 3.93e9T^{2} \)
89 \( 1 - 5.83e4T + 5.58e9T^{2} \)
97 \( 1 - 6.18e4iT - 8.58e9T^{2} \)
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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

−12.15534914665792471457925463811, −10.36266958612435909536693773626, −9.772361688845320655398354773084, −8.097852980697588321395724344927, −7.63841744785462485545660176467, −7.10534771703140453932142865142, −5.55825381122501151466889487914, −4.72093516145011337267129586411, −3.55784047642722599052721239434, −0.51798946305036657908056985984, 0.56139382193985278651182027285, 2.55289407101462069114670121358, 2.84073090600592660758717720514, 4.63268867216652392293795095881, 5.30175802125357860549781638329, 7.57320150122575645532794188946, 8.578363813576860815314742337791, 9.604116961983882241752805666707, 10.32180099994886731255701658093, 11.54422201717606705176422132906

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