Properties

Label 2-3-3.2-c66-0-17
Degree $2$
Conductor $3$
Sign $0.683 - 0.729i$
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  − 1.68e10i·2-s + (−3.79e15 + 4.05e15i)3-s − 2.09e20·4-s + 9.39e22i·5-s + (6.83e25 + 6.39e25i)6-s − 5.61e27·7-s + 2.29e30i·8-s + (−2.02e30 − 3.08e31i)9-s + 1.58e33·10-s − 3.11e34i·11-s + (7.97e35 − 8.51e35i)12-s + 6.10e35·13-s + 9.46e37i·14-s + (−3.81e38 − 3.56e38i)15-s + 2.31e40·16-s − 4.49e40i·17-s + ⋯
L(s)  = 1  − 1.96i·2-s + (−0.683 + 0.729i)3-s − 2.84·4-s + 0.806i·5-s + (1.43 + 1.34i)6-s − 0.726·7-s + 3.61i·8-s + (−0.0656 − 0.997i)9-s + 1.58·10-s − 1.34i·11-s + (1.94 − 2.07i)12-s + 0.106·13-s + 1.42i·14-s + (−0.588 − 0.551i)15-s + 4.24·16-s − 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.683 - 0.729i$
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: \((2,\ 3,\ (\ :33),\ 0.683 - 0.729i)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(0.07271898642\)
\(L(\frac12)\) \(\approx\) \(0.07271898642\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.79e15 - 4.05e15i)T \)
good2 \( 1 + 1.68e10iT - 7.37e19T^{2} \)
5 \( 1 - 9.39e22iT - 1.35e46T^{2} \)
7 \( 1 + 5.61e27T + 5.97e55T^{2} \)
11 \( 1 + 3.11e34iT - 5.39e68T^{2} \)
13 \( 1 - 6.10e35T + 3.31e73T^{2} \)
17 \( 1 + 4.49e40iT - 1.62e81T^{2} \)
19 \( 1 + 1.04e42T + 2.49e84T^{2} \)
23 \( 1 + 9.46e44iT - 7.48e89T^{2} \)
29 \( 1 + 8.84e47iT - 3.29e96T^{2} \)
31 \( 1 + 2.69e48T + 2.69e98T^{2} \)
37 \( 1 - 8.18e51T + 3.17e103T^{2} \)
41 \( 1 - 4.96e52iT - 2.77e106T^{2} \)
43 \( 1 + 5.17e53T + 6.44e107T^{2} \)
47 \( 1 + 5.70e54iT - 2.28e110T^{2} \)
53 \( 1 - 7.48e56iT - 6.34e113T^{2} \)
59 \( 1 + 1.33e58iT - 7.52e116T^{2} \)
61 \( 1 - 1.11e59T + 6.78e117T^{2} \)
67 \( 1 + 1.40e60T + 3.31e120T^{2} \)
71 \( 1 + 1.22e61iT - 1.52e122T^{2} \)
73 \( 1 + 4.81e61T + 9.53e122T^{2} \)
79 \( 1 + 1.69e62T + 1.75e125T^{2} \)
83 \( 1 + 3.17e63iT - 4.56e126T^{2} \)
89 \( 1 + 1.06e64iT - 4.56e128T^{2} \)
97 \( 1 + 2.41e65T + 1.33e131T^{2} \)
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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

−11.66662161688837079891554430792, −10.87342728984535434425290054093, −9.973944457983683932441326890216, −8.815508870576443308718431650978, −6.08438554891006513720144119917, −4.58616844897631245499381515709, −3.37750727892320146373719401023, −2.69658004091564736820522756324, −0.75664464536897475516196604176, −0.02984709106832242737635492939, 1.30410735405108834220690267862, 4.21986805724629443566560500012, 5.29518671558217532393898516131, 6.34361351413923426483434139228, 7.29389188699405955545348327572, 8.466558554800520506971334734868, 9.832759466263209903631652601110, 12.65534442911164976062557680386, 13.13104267069216199492871589228, 14.86559542375743540956270580209

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