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Properties

Label 2-3-3.2-c64-0-13
Degree $2$
Conductor $3$
Sign $0.458 + 0.888i$
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  − 9.53e8i·2-s + (8.49e14 + 1.64e15i)3-s + 1.75e19·4-s − 2.45e22i·5-s + (1.57e24 − 8.10e23i)6-s − 1.16e27·7-s − 3.43e28i·8-s + (−1.98e30 + 2.79e30i)9-s − 2.34e31·10-s + 6.76e32i·11-s + (1.49e34 + 2.88e34i)12-s − 2.73e35·13-s + 1.10e36i·14-s + (4.04e37 − 2.08e37i)15-s + 2.90e38·16-s + 1.54e39i·17-s + ⋯
L(s)  = 1  − 0.222i·2-s + (0.458 + 0.888i)3-s + 0.950·4-s − 1.05i·5-s + (0.197 − 0.101i)6-s − 1.05·7-s − 0.433i·8-s + (−0.579 + 0.814i)9-s − 0.234·10-s + 0.320i·11-s + (0.435 + 0.844i)12-s − 0.617·13-s + 0.233i·14-s + (0.936 − 0.483i)15-s + 0.854·16-s + 0.652i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.458 + 0.888i$
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: \((2,\ 3,\ (\ :32),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(2.440488937\)
\(L(\frac12)\) \(\approx\) \(2.440488937\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.49e14 - 1.64e15i)T \)
good2 \( 1 + 9.53e8iT - 1.84e19T^{2} \)
5 \( 1 + 2.45e22iT - 5.42e44T^{2} \)
7 \( 1 + 1.16e27T + 1.21e54T^{2} \)
11 \( 1 - 6.76e32iT - 4.45e66T^{2} \)
13 \( 1 + 2.73e35T + 1.96e71T^{2} \)
17 \( 1 - 1.54e39iT - 5.60e78T^{2} \)
19 \( 1 - 8.35e40T + 6.92e81T^{2} \)
23 \( 1 + 2.80e43iT - 1.41e87T^{2} \)
29 \( 1 + 9.86e46iT - 3.92e93T^{2} \)
31 \( 1 - 2.10e47T + 2.79e95T^{2} \)
37 \( 1 - 2.82e50T + 2.31e100T^{2} \)
41 \( 1 + 1.43e51iT - 1.65e103T^{2} \)
43 \( 1 - 4.26e51T + 3.48e104T^{2} \)
47 \( 1 + 2.67e53iT - 1.03e107T^{2} \)
53 \( 1 + 2.54e55iT - 2.25e110T^{2} \)
59 \( 1 + 2.49e55iT - 2.16e113T^{2} \)
61 \( 1 + 2.06e57T + 1.82e114T^{2} \)
67 \( 1 + 2.49e58T + 7.39e116T^{2} \)
71 \( 1 + 3.22e59iT - 3.02e118T^{2} \)
73 \( 1 + 3.78e59T + 1.78e119T^{2} \)
79 \( 1 - 1.02e61T + 2.80e121T^{2} \)
83 \( 1 + 3.30e61iT - 6.62e122T^{2} \)
89 \( 1 + 3.88e62iT - 5.76e124T^{2} \)
97 \( 1 + 5.25e62T + 1.42e127T^{2} \)
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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

−13.11948655075852823309261079752, −11.89373094312334364862011800396, −10.24338676563365655733902876246, −9.353350849789898177628116616966, −7.81767254819371243704503166519, −6.09663765696843246143993142485, −4.60867143418969474970339690616, −3.28546838382608487609906253596, −2.16281351493931578748633831735, −0.53175771890721291192394936705, 1.15786069470337618667368556060, 2.78343354073021905852734871215, 3.06217178481273803981321126063, 5.91408293799318691014114942684, 6.89890028284760989967334213548, 7.58788947973015610094726856909, 9.538289445401522136054769402484, 11.13182729660792317671607469741, 12.37569869512652209188426767828, 13.90608434795378401575326625431

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