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Properties

Label 2-3-3.2-c24-0-0
Degree $2$
Conductor $3$
Sign $0.304 + 0.952i$
Analytic cond. $10.9490$
Root an. cond. $3.30892$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.84e3i·2-s + (1.62e5 + 5.06e5i)3-s − 3.00e7·4-s − 3.79e8i·5-s + (−3.46e9 + 1.10e9i)6-s − 1.14e10·7-s − 9.08e10i·8-s + (−2.29e11 + 1.63e11i)9-s + 2.59e12·10-s − 1.05e12i·11-s + (−4.86e12 − 1.52e13i)12-s − 1.55e13·13-s − 7.81e13i·14-s + (1.91e14 − 6.14e13i)15-s + 1.17e14·16-s + 7.28e14i·17-s + ⋯
L(s)  = 1  + 1.67i·2-s + (0.304 + 0.952i)3-s − 1.79·4-s − 1.55i·5-s + (−1.59 + 0.509i)6-s − 0.824·7-s − 1.32i·8-s + (−0.814 + 0.580i)9-s + 2.59·10-s − 0.336i·11-s + (−0.546 − 1.70i)12-s − 0.666·13-s − 1.37i·14-s + (1.47 − 0.473i)15-s + 0.418·16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.304 + 0.952i$
Analytic conductor: \(10.9490\)
Root analytic conductor: \(3.30892\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :12),\ 0.304 + 0.952i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.233832 - 0.170673i\)
\(L(\frac12)\) \(\approx\) \(0.233832 - 0.170673i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62e5 - 5.06e5i)T \)
good2 \( 1 - 6.84e3iT - 1.67e7T^{2} \)
5 \( 1 + 3.79e8iT - 5.96e16T^{2} \)
7 \( 1 + 1.14e10T + 1.91e20T^{2} \)
11 \( 1 + 1.05e12iT - 9.84e24T^{2} \)
13 \( 1 + 1.55e13T + 5.42e26T^{2} \)
17 \( 1 - 7.28e14iT - 3.39e29T^{2} \)
19 \( 1 + 1.02e15T + 4.89e30T^{2} \)
23 \( 1 - 4.33e15iT - 4.80e32T^{2} \)
29 \( 1 + 1.40e17iT - 1.25e35T^{2} \)
31 \( 1 + 9.18e17T + 6.20e35T^{2} \)
37 \( 1 - 1.09e19T + 4.33e37T^{2} \)
41 \( 1 - 1.60e19iT - 5.09e38T^{2} \)
43 \( 1 + 3.62e19T + 1.59e39T^{2} \)
47 \( 1 + 4.21e19iT - 1.35e40T^{2} \)
53 \( 1 - 5.22e20iT - 2.41e41T^{2} \)
59 \( 1 - 2.40e21iT - 3.16e42T^{2} \)
61 \( 1 - 8.29e20T + 7.04e42T^{2} \)
67 \( 1 - 9.12e20T + 6.69e43T^{2} \)
71 \( 1 + 5.88e21iT - 2.69e44T^{2} \)
73 \( 1 - 1.41e22T + 5.24e44T^{2} \)
79 \( 1 + 1.14e23T + 3.49e45T^{2} \)
83 \( 1 + 4.90e22iT - 1.14e46T^{2} \)
89 \( 1 - 3.49e23iT - 6.10e46T^{2} \)
97 \( 1 + 9.80e23T + 4.81e47T^{2} \)
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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

−21.74730204554545569887324133960, −19.85510057191723153412299193503, −16.96010183253757644806061842911, −16.34846081388305148050827083589, −14.99688025289219396064097793128, −13.12156359925170713156463831347, −9.431985804039432003731389419273, −8.261846957802239371978399847218, −5.74211752795144848573644770440, −4.32693866840688886923668904176, 0.12072612718175976728327597425, 2.27006090983739967173445177654, 3.21506201222435282932798577571, 6.94938269562008727452259251303, 9.712623886490538871507430587072, 11.34926088893946540977477799754, 12.85210962373225076667747579903, 14.38199923082447249538132874794, 18.12864979055082830321382711245, 18.97688350514255949950040698354

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