Properties

Label 2-3-3.2-c66-0-15
Degree $2$
Conductor $3$
Sign $-0.950 + 0.311i$
Analytic cond. $82.7604$
Root an. cond. $9.09727$
Motivic weight $66$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.59e10i·2-s + (5.28e15 − 1.73e15i)3-s − 1.81e20·4-s − 8.56e22i·5-s + (−2.76e25 − 8.44e25i)6-s + 1.24e28·7-s + 1.72e30i·8-s + (2.48e31 − 1.83e31i)9-s − 1.36e33·10-s + 3.23e34i·11-s + (−9.59e35 + 3.14e35i)12-s + 4.92e36·13-s − 1.99e38i·14-s + (−1.48e38 − 4.52e38i)15-s + 1.41e40·16-s + 1.87e40i·17-s + ⋯
L(s)  = 1  − 1.86i·2-s + (0.950 − 0.311i)3-s − 2.46·4-s − 0.736i·5-s + (−0.579 − 1.76i)6-s + 1.61·7-s + 2.71i·8-s + (0.805 − 0.592i)9-s − 1.36·10-s + 1.39i·11-s + (−2.33 + 0.767i)12-s + 0.855·13-s − 2.99i·14-s + (−0.229 − 0.699i)15-s + 2.59·16-s + 0.466i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(82.7604\)
Root analytic conductor: \(9.09727\)
Motivic weight: \(66\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :33),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(3.822930194\)
\(L(\frac12)\) \(\approx\) \(3.822930194\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.28e15 + 1.73e15i)T \)
good2 \( 1 + 1.59e10iT - 7.37e19T^{2} \)
5 \( 1 + 8.56e22iT - 1.35e46T^{2} \)
7 \( 1 - 1.24e28T + 5.97e55T^{2} \)
11 \( 1 - 3.23e34iT - 5.39e68T^{2} \)
13 \( 1 - 4.92e36T + 3.31e73T^{2} \)
17 \( 1 - 1.87e40iT - 1.62e81T^{2} \)
19 \( 1 - 1.55e42T + 2.49e84T^{2} \)
23 \( 1 + 2.71e44iT - 7.48e89T^{2} \)
29 \( 1 + 1.13e48iT - 3.29e96T^{2} \)
31 \( 1 - 1.24e49T + 2.69e98T^{2} \)
37 \( 1 - 3.41e51T + 3.17e103T^{2} \)
41 \( 1 - 2.61e53iT - 2.77e106T^{2} \)
43 \( 1 - 4.27e53T + 6.44e107T^{2} \)
47 \( 1 + 1.98e55iT - 2.28e110T^{2} \)
53 \( 1 - 6.52e55iT - 6.34e113T^{2} \)
59 \( 1 + 6.23e57iT - 7.52e116T^{2} \)
61 \( 1 + 9.12e58T + 6.78e117T^{2} \)
67 \( 1 + 1.10e60T + 3.31e120T^{2} \)
71 \( 1 - 2.12e61iT - 1.52e122T^{2} \)
73 \( 1 + 5.91e60T + 9.53e122T^{2} \)
79 \( 1 + 3.17e62T + 1.75e125T^{2} \)
83 \( 1 + 1.29e63iT - 4.56e126T^{2} \)
89 \( 1 - 1.78e64iT - 4.56e128T^{2} \)
97 \( 1 + 5.99e65T + 1.33e131T^{2} \)
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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

−12.56755259597534659845749321790, −11.52570662491442693723275363569, −10.03074008939117264593646526265, −8.835126224927428723282915368631, −7.926234834445092419589188766863, −4.83078387296812571019571951436, −4.10166177898838442691035526402, −2.56850632896400153585875678978, −1.51750962398846523926953263636, −1.09225795513881845786939043028, 1.09204184071167648624280423727, 3.21005551193221731656620758725, 4.55026325588218883893959581635, 5.74322033547277182926536262037, 7.29406696916093669683625302572, 8.164740608707532310137200213189, 9.021271352239536303440498341881, 10.91367388172106299471969112763, 13.81647050798447723853857753894, 14.13843161501696777823263458833

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