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Properties

Label 2-3-3.2-c64-0-9
Degree $2$
Conductor $3$
Sign $0.194 + 0.980i$
Analytic cond. $77.8210$
Root an. cond. $8.82162$
Motivic weight $64$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.99e9i·2-s + (3.59e14 + 1.81e15i)3-s − 3.05e19·4-s − 1.06e22i·5-s + (1.27e25 − 2.51e24i)6-s + 9.75e26·7-s + 8.45e28i·8-s + (−3.17e30 + 1.30e30i)9-s − 7.46e31·10-s − 1.03e33i·11-s + (−1.09e34 − 5.54e34i)12-s + 5.97e35·13-s − 6.82e36i·14-s + (1.94e37 − 3.83e36i)15-s + 2.83e37·16-s + 4.93e38i·17-s + ⋯
L(s)  = 1  − 1.62i·2-s + (0.194 + 0.980i)3-s − 1.65·4-s − 0.458i·5-s + (1.59 − 0.316i)6-s + 0.883·7-s + 1.06i·8-s + (−0.924 + 0.380i)9-s − 0.746·10-s − 0.489i·11-s + (−0.321 − 1.62i)12-s + 1.35·13-s − 1.43i·14-s + (0.449 − 0.0890i)15-s + 0.0832·16-s + 0.208i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(77.8210\)
Root analytic conductor: \(8.82162\)
Motivic weight: \(64\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :32),\ 0.194 + 0.980i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(2.372127295\)
\(L(\frac12)\) \(\approx\) \(2.372127295\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.59e14 - 1.81e15i)T \)
good2 \( 1 + 6.99e9iT - 1.84e19T^{2} \)
5 \( 1 + 1.06e22iT - 5.42e44T^{2} \)
7 \( 1 - 9.75e26T + 1.21e54T^{2} \)
11 \( 1 + 1.03e33iT - 4.45e66T^{2} \)
13 \( 1 - 5.97e35T + 1.96e71T^{2} \)
17 \( 1 - 4.93e38iT - 5.60e78T^{2} \)
19 \( 1 + 6.78e40T + 6.92e81T^{2} \)
23 \( 1 - 6.53e43iT - 1.41e87T^{2} \)
29 \( 1 - 5.48e46iT - 3.92e93T^{2} \)
31 \( 1 + 4.13e47T + 2.79e95T^{2} \)
37 \( 1 - 1.79e50T + 2.31e100T^{2} \)
41 \( 1 + 5.76e51iT - 1.65e103T^{2} \)
43 \( 1 - 3.15e52T + 3.48e104T^{2} \)
47 \( 1 - 3.13e53iT - 1.03e107T^{2} \)
53 \( 1 + 1.48e55iT - 2.25e110T^{2} \)
59 \( 1 + 4.53e56iT - 2.16e113T^{2} \)
61 \( 1 - 2.37e57T + 1.82e114T^{2} \)
67 \( 1 - 1.24e58T + 7.39e116T^{2} \)
71 \( 1 - 8.24e58iT - 3.02e118T^{2} \)
73 \( 1 + 9.65e58T + 1.78e119T^{2} \)
79 \( 1 - 1.50e60T + 2.80e121T^{2} \)
83 \( 1 + 3.18e61iT - 6.62e122T^{2} \)
89 \( 1 - 2.95e62iT - 5.76e124T^{2} \)
97 \( 1 + 8.64e62T + 1.42e127T^{2} \)
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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

−12.97673776088182334786409618220, −11.29687988643259639871623309890, −10.81952264725994217827901759383, −9.283828522353299623300168217894, −8.411284754736892554961606910397, −5.45080823610179520734194466354, −4.20232767650387185707591446262, −3.36226189015623200623791563532, −1.90717682168638852731399618790, −0.820920773988485690420255987320, 0.78689208089837537347389039706, 2.33989991350757249297660041654, 4.39190335702643860009937590082, 5.94381691125782034211880750842, 6.79168683399754033377515087371, 7.936053611584844626740508573873, 8.751449204479523584120947074639, 11.12294069645863904329377119797, 12.98073519213179486242570211732, 14.26603958324035981009380762183

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