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Properties

Label 2-3-3.2-c70-0-10
Degree $2$
Conductor $3$
Sign $-0.473 - 0.880i$
Analytic cond. $93.0951$
Root an. cond. $9.64858$
Motivic weight $70$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.63e10i·2-s + (2.36e16 + 4.40e16i)3-s − 3.22e21·4-s − 6.32e23i·5-s + (−2.92e27 + 1.57e27i)6-s − 2.54e29·7-s − 1.35e32i·8-s + (−1.38e33 + 2.08e33i)9-s + 4.20e34·10-s + 1.33e36i·11-s + (−7.64e37 − 1.42e38i)12-s − 1.53e39·13-s − 1.69e40i·14-s + (2.78e40 − 1.49e40i)15-s + 5.20e42·16-s − 7.59e42i·17-s + ⋯
L(s)  = 1  + 1.93i·2-s + (0.473 + 0.880i)3-s − 2.73·4-s − 0.217i·5-s + (−1.70 + 0.915i)6-s − 0.672·7-s − 3.34i·8-s + (−0.551 + 0.834i)9-s + 0.420·10-s + 0.476i·11-s + (−1.29 − 2.40i)12-s − 1.57·13-s − 1.29i·14-s + (0.191 − 0.102i)15-s + 3.73·16-s − 0.652i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(93.0951\)
Root analytic conductor: \(9.64858\)
Motivic weight: \(70\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :35),\ -0.473 - 0.880i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(1.011303370\)
\(L(\frac12)\) \(\approx\) \(1.011303370\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.36e16 - 4.40e16i)T \)
good2 \( 1 - 6.63e10iT - 1.18e21T^{2} \)
5 \( 1 + 6.32e23iT - 8.47e48T^{2} \)
7 \( 1 + 2.54e29T + 1.43e59T^{2} \)
11 \( 1 - 1.33e36iT - 7.89e72T^{2} \)
13 \( 1 + 1.53e39T + 9.46e77T^{2} \)
17 \( 1 + 7.59e42iT - 1.35e86T^{2} \)
19 \( 1 - 8.19e44T + 3.25e89T^{2} \)
23 \( 1 + 6.91e47iT - 2.09e95T^{2} \)
29 \( 1 - 4.05e50iT - 2.33e102T^{2} \)
31 \( 1 + 1.63e52T + 2.48e104T^{2} \)
37 \( 1 - 3.41e54T + 5.94e109T^{2} \)
41 \( 1 - 4.15e56iT - 7.85e112T^{2} \)
43 \( 1 + 8.83e56T + 2.20e114T^{2} \)
47 \( 1 - 4.77e57iT - 1.11e117T^{2} \)
53 \( 1 - 5.62e59iT - 5.00e120T^{2} \)
59 \( 1 + 9.59e61iT - 9.11e123T^{2} \)
61 \( 1 + 8.18e61T + 9.39e124T^{2} \)
67 \( 1 - 2.40e63T + 6.68e127T^{2} \)
71 \( 1 + 7.14e64iT - 3.87e129T^{2} \)
73 \( 1 - 1.52e65T + 2.70e130T^{2} \)
79 \( 1 + 2.33e66T + 6.82e132T^{2} \)
83 \( 1 - 2.67e67iT - 2.16e134T^{2} \)
89 \( 1 + 5.29e67iT - 2.86e136T^{2} \)
97 \( 1 - 4.21e69T + 1.18e139T^{2} \)
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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

−14.19657852957955930214770444903, −12.73905573807368135918460780173, −9.857020255237730720517635993770, −9.242401538211983980660008234129, −7.86510341776379346813146891392, −6.80892863652547116372659279159, −5.20243204886437460892097505780, −4.59002895921072235975545464713, −3.05236678321904749459184124749, −0.32670744965003706152252734224, 0.69550468360578583956508595161, 1.81084540156515777760123998195, 2.87044211441959171345317899066, 3.58402138022120215481124425766, 5.40038986924038966506735229042, 7.48320714933412573861391016060, 9.038057644822097083390948487684, 9.954436964166385504138003270416, 11.50744058718253661229074200447, 12.46205956030255175735016817764

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