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Properties

Label 2-3-3.2-c64-0-11
Degree $2$
Conductor $3$
Sign $0.976 + 0.217i$
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  + 6.95e9i·2-s + (1.80e15 + 4.02e14i)3-s − 2.99e19·4-s + 4.26e21i·5-s + (−2.80e24 + 1.25e25i)6-s − 1.59e27·7-s − 7.98e28i·8-s + (3.10e30 + 1.45e30i)9-s − 2.96e31·10-s + 1.95e33i·11-s + (−5.41e34 − 1.20e34i)12-s − 4.09e35·13-s − 1.10e37i·14-s + (−1.71e36 + 7.71e36i)15-s + 3.09e36·16-s − 1.60e39i·17-s + ⋯
L(s)  = 1  + 1.61i·2-s + (0.976 + 0.217i)3-s − 1.62·4-s + 0.183i·5-s + (−0.352 + 1.58i)6-s − 1.44·7-s − 1.00i·8-s + (0.905 + 0.424i)9-s − 0.296·10-s + 0.926i·11-s + (−1.58 − 0.352i)12-s − 0.924·13-s − 2.33i·14-s + (−0.0398 + 0.178i)15-s + 0.00909·16-s − 0.677i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.976 + 0.217i$
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: \((2,\ 3,\ (\ :32),\ 0.976 + 0.217i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(0.03727536069\)
\(L(\frac12)\) \(\approx\) \(0.03727536069\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.80e15 - 4.02e14i)T \)
good2 \( 1 - 6.95e9iT - 1.84e19T^{2} \)
5 \( 1 - 4.26e21iT - 5.42e44T^{2} \)
7 \( 1 + 1.59e27T + 1.21e54T^{2} \)
11 \( 1 - 1.95e33iT - 4.45e66T^{2} \)
13 \( 1 + 4.09e35T + 1.96e71T^{2} \)
17 \( 1 + 1.60e39iT - 5.60e78T^{2} \)
19 \( 1 + 4.25e40T + 6.92e81T^{2} \)
23 \( 1 - 6.67e43iT - 1.41e87T^{2} \)
29 \( 1 + 3.89e46iT - 3.92e93T^{2} \)
31 \( 1 + 3.52e47T + 2.79e95T^{2} \)
37 \( 1 + 1.20e49T + 2.31e100T^{2} \)
41 \( 1 + 7.11e51iT - 1.65e103T^{2} \)
43 \( 1 - 1.24e52T + 3.48e104T^{2} \)
47 \( 1 + 5.35e53iT - 1.03e107T^{2} \)
53 \( 1 - 1.42e55iT - 2.25e110T^{2} \)
59 \( 1 + 6.80e56iT - 2.16e113T^{2} \)
61 \( 1 + 3.82e56T + 1.82e114T^{2} \)
67 \( 1 - 4.35e58T + 7.39e116T^{2} \)
71 \( 1 - 6.48e58iT - 3.02e118T^{2} \)
73 \( 1 - 1.90e59T + 1.78e119T^{2} \)
79 \( 1 + 9.64e60T + 2.80e121T^{2} \)
83 \( 1 + 1.66e61iT - 6.62e122T^{2} \)
89 \( 1 - 1.95e62iT - 5.76e124T^{2} \)
97 \( 1 + 2.35e63T + 1.42e127T^{2} \)
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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

−13.94745862657625573946269367550, −12.76076822855299631166741988743, −9.862712893878846205769245596769, −9.081916950921044478960398425574, −7.46378076627629470288927097468, −6.85630913963398181370245744928, −5.21974594989533944856693389591, −3.80705181869510371408922131680, −2.38263686539566456406770565286, −0.008047717372634555662670013002, 1.08414060232827837735028601714, 2.53675548172439830029045933768, 3.14097754529863385867413070158, 4.30547256319272023373124520594, 6.59986451247626350559579243767, 8.578822662519663530836051181361, 9.571779054048542876558865413537, 10.63588974392408999709441289921, 12.56999123656918738536854528755, 12.94678399948465378458719684673

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