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Properties

Label 2-3-3.2-c42-0-4
Degree $2$
Conductor $3$
Sign $-0.977 - 0.210i$
Analytic cond. $33.5183$
Root an. cond. $5.78950$
Motivic weight $42$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.75e6i·2-s + (1.02e10 + 2.19e9i)3-s − 9.68e12·4-s − 5.14e14i·5-s + (−8.24e15 + 3.83e16i)6-s + 7.24e17·7-s − 1.98e19i·8-s + (9.97e19 + 4.49e19i)9-s + 1.93e21·10-s + 5.80e21i·11-s + (−9.90e22 − 2.12e22i)12-s − 2.70e23·13-s + 2.71e24i·14-s + (1.13e24 − 5.25e24i)15-s + 3.19e25·16-s + 9.64e25i·17-s + ⋯
L(s)  = 1  + 1.78i·2-s + (0.977 + 0.210i)3-s − 2.20·4-s − 1.07i·5-s + (−0.375 + 1.74i)6-s + 1.29·7-s − 2.15i·8-s + (0.911 + 0.410i)9-s + 1.93·10-s + 0.784i·11-s + (−2.15 − 0.462i)12-s − 1.09·13-s + 2.32i·14-s + (0.226 − 1.05i)15-s + 1.65·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(33.5183\)
Root analytic conductor: \(5.78950\)
Motivic weight: \(42\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :21),\ -0.977 - 0.210i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(2.695663742\)
\(L(\frac12)\) \(\approx\) \(2.695663742\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.02e10 - 2.19e9i)T \)
good2 \( 1 - 3.75e6iT - 4.39e12T^{2} \)
5 \( 1 + 5.14e14iT - 2.27e29T^{2} \)
7 \( 1 - 7.24e17T + 3.11e35T^{2} \)
11 \( 1 - 5.80e21iT - 5.47e43T^{2} \)
13 \( 1 + 2.70e23T + 6.10e46T^{2} \)
17 \( 1 - 9.64e25iT - 4.77e51T^{2} \)
19 \( 1 - 4.61e26T + 5.10e53T^{2} \)
23 \( 1 - 3.87e28iT - 1.55e57T^{2} \)
29 \( 1 - 4.99e30iT - 2.63e61T^{2} \)
31 \( 1 + 6.62e30T + 4.33e62T^{2} \)
37 \( 1 - 1.25e33T + 7.31e65T^{2} \)
41 \( 1 - 6.68e33iT - 5.45e67T^{2} \)
43 \( 1 - 1.15e34T + 4.03e68T^{2} \)
47 \( 1 + 6.11e34iT - 1.69e70T^{2} \)
53 \( 1 + 1.29e36iT - 2.62e72T^{2} \)
59 \( 1 + 6.95e36iT - 2.37e74T^{2} \)
61 \( 1 + 4.26e37T + 9.63e74T^{2} \)
67 \( 1 + 1.36e38T + 4.95e76T^{2} \)
71 \( 1 - 7.56e38iT - 5.66e77T^{2} \)
73 \( 1 - 7.90e38T + 1.81e78T^{2} \)
79 \( 1 + 1.56e38T + 5.01e79T^{2} \)
83 \( 1 - 2.41e40iT - 3.99e80T^{2} \)
89 \( 1 + 1.04e41iT - 7.48e81T^{2} \)
97 \( 1 - 3.83e41T + 2.78e83T^{2} \)
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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

−16.70881113589277059568314612530, −15.18449048448021277631535427996, −14.41301786348410092911618841287, −12.85943484082784398595176016185, −9.479744847294052402926254237254, −8.288740091451856701143249874191, −7.45955693210022903325902072110, −5.17207589671405116658522715874, −4.36710100779523950640896075575, −1.51217317833512927964915460531, 0.78613974847351663536532908469, 2.29049787614503418150105802966, 2.96511028068308101683535679188, 4.54403831364040380289587996412, 7.66823179979168807897408164306, 9.309900124241561168184742458765, 10.73334854004425740666963008939, 11.92326345525225654935099405361, 13.78459629945893307285622573386, 14.58711914261934392867622660129

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