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Properties

Label 2-3-3.2-c64-0-2
Degree $2$
Conductor $3$
Sign $0.301 + 0.953i$
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  + 4.03e9i·2-s + (5.57e14 + 1.76e15i)3-s + 2.19e18·4-s + 2.91e22i·5-s + (−7.12e24 + 2.24e24i)6-s + 6.92e26·7-s + 8.32e28i·8-s + (−2.81e30 + 1.97e30i)9-s − 1.17e32·10-s − 2.05e33i·11-s + (1.22e33 + 3.87e33i)12-s − 9.38e34·13-s + 2.79e36i·14-s + (−5.15e37 + 1.62e37i)15-s − 2.95e38·16-s − 1.29e39i·17-s + ⋯
L(s)  = 1  + 0.938i·2-s + (0.301 + 0.953i)3-s + 0.118·4-s + 1.25i·5-s + (−0.895 + 0.282i)6-s + 0.626·7-s + 1.05i·8-s + (−0.818 + 0.574i)9-s − 1.17·10-s − 0.972i·11-s + (0.0357 + 0.113i)12-s − 0.212·13-s + 0.588i·14-s + (−1.19 + 0.377i)15-s − 0.867·16-s − 0.547i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.301 + 0.953i$
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: \((2,\ 3,\ (\ :32),\ 0.301 + 0.953i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(1.214553926\)
\(L(\frac12)\) \(\approx\) \(1.214553926\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.57e14 - 1.76e15i)T \)
good2 \( 1 - 4.03e9iT - 1.84e19T^{2} \)
5 \( 1 - 2.91e22iT - 5.42e44T^{2} \)
7 \( 1 - 6.92e26T + 1.21e54T^{2} \)
11 \( 1 + 2.05e33iT - 4.45e66T^{2} \)
13 \( 1 + 9.38e34T + 1.96e71T^{2} \)
17 \( 1 + 1.29e39iT - 5.60e78T^{2} \)
19 \( 1 + 2.23e40T + 6.92e81T^{2} \)
23 \( 1 + 4.35e43iT - 1.41e87T^{2} \)
29 \( 1 - 1.55e45iT - 3.92e93T^{2} \)
31 \( 1 + 9.94e47T + 2.79e95T^{2} \)
37 \( 1 + 6.98e49T + 2.31e100T^{2} \)
41 \( 1 + 4.06e51iT - 1.65e103T^{2} \)
43 \( 1 + 3.69e52T + 3.48e104T^{2} \)
47 \( 1 - 5.97e53iT - 1.03e107T^{2} \)
53 \( 1 - 1.81e55iT - 2.25e110T^{2} \)
59 \( 1 - 1.74e56iT - 2.16e113T^{2} \)
61 \( 1 + 1.21e57T + 1.82e114T^{2} \)
67 \( 1 - 4.80e58T + 7.39e116T^{2} \)
71 \( 1 - 4.77e58iT - 3.02e118T^{2} \)
73 \( 1 - 3.60e59T + 1.78e119T^{2} \)
79 \( 1 + 3.79e60T + 2.80e121T^{2} \)
83 \( 1 + 2.93e61iT - 6.62e122T^{2} \)
89 \( 1 + 1.74e62iT - 5.76e124T^{2} \)
97 \( 1 - 6.41e63T + 1.42e127T^{2} \)
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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

−14.68207809491848713575266565931, −14.17857818012284226581916352852, −11.31032109210793885667257054246, −10.60615098346967452577703654412, −8.758082328386285405753572136826, −7.53826944313510917048843595418, −6.23525320777950670978133600880, −4.99352383262285461593794474986, −3.32563414315676380226959977437, −2.27326781382121391982407964822, 0.23374673988039799507338747074, 1.60107998100754687691625013724, 1.84409746913275247782371284123, 3.60327247440411547736856569836, 5.15935260961632437095033602025, 6.95919455777052089311390404155, 8.264110253034158564119240168529, 9.614634932137486633914370593438, 11.41972161892687600979110511041, 12.43063140885959766918076896165

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