Properties

Label 2-3-3.2-c34-0-7
Degree $2$
Conductor $3$
Sign $-0.510 + 0.859i$
Analytic cond. $21.9676$
Root an. cond. $4.68697$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.52e5i·2-s + (6.59e7 − 1.11e8i)3-s − 6.06e9·4-s + 1.02e12i·5-s + (−1.69e13 − 1.00e13i)6-s + 3.85e14·7-s − 1.69e15i·8-s + (−7.96e15 − 1.46e16i)9-s + 1.56e17·10-s − 1.72e17i·11-s + (−4.00e17 + 6.73e17i)12-s + 1.38e19·13-s − 5.88e19i·14-s + (1.13e20 + 6.75e19i)15-s − 3.62e20·16-s + 1.27e20i·17-s + ⋯
L(s)  = 1  − 1.16i·2-s + (0.510 − 0.859i)3-s − 0.352·4-s + 1.34i·5-s + (−0.999 − 0.594i)6-s + 1.65·7-s − 0.752i·8-s + (−0.477 − 0.878i)9-s + 1.56·10-s − 0.342i·11-s + (−0.180 + 0.303i)12-s + 1.60·13-s − 1.92i·14-s + (1.15 + 0.685i)15-s − 1.22·16-s + 0.154i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(21.9676\)
Root analytic conductor: \(4.68697\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :17),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.48427 - 2.60906i\)
\(L(\frac12)\) \(\approx\) \(1.48427 - 2.60906i\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.59e7 + 1.11e8i)T \)
good2 \( 1 + 1.52e5iT - 1.71e10T^{2} \)
5 \( 1 - 1.02e12iT - 5.82e23T^{2} \)
7 \( 1 - 3.85e14T + 5.41e28T^{2} \)
11 \( 1 + 1.72e17iT - 2.55e35T^{2} \)
13 \( 1 - 1.38e19T + 7.48e37T^{2} \)
17 \( 1 - 1.27e20iT - 6.84e41T^{2} \)
19 \( 1 + 1.48e21T + 3.00e43T^{2} \)
23 \( 1 + 7.90e22iT - 1.98e46T^{2} \)
29 \( 1 - 3.64e23iT - 5.26e49T^{2} \)
31 \( 1 - 6.92e24T + 5.08e50T^{2} \)
37 \( 1 - 5.11e25T + 2.08e53T^{2} \)
41 \( 1 - 2.34e27iT - 6.83e54T^{2} \)
43 \( 1 + 8.17e27T + 3.45e55T^{2} \)
47 \( 1 + 4.43e28iT - 7.10e56T^{2} \)
53 \( 1 - 1.03e29iT - 4.22e58T^{2} \)
59 \( 1 + 8.33e29iT - 1.61e60T^{2} \)
61 \( 1 - 6.58e29T + 5.02e60T^{2} \)
67 \( 1 - 1.06e31T + 1.22e62T^{2} \)
71 \( 1 - 5.27e30iT - 8.76e62T^{2} \)
73 \( 1 + 4.83e31T + 2.25e63T^{2} \)
79 \( 1 - 2.88e32T + 3.30e64T^{2} \)
83 \( 1 + 3.03e31iT - 1.77e65T^{2} \)
89 \( 1 - 1.20e33iT - 1.90e66T^{2} \)
97 \( 1 + 4.59e33T + 3.55e67T^{2} \)
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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

−18.12651067714925502577894601878, −14.88916326930881875552974478323, −13.58601859432386173462966354278, −11.61264012952392618457478007287, −10.76673159227554532273947569793, −8.296129341847261408016927754297, −6.59476206907695769167298313196, −3.56375999125113166625997027995, −2.24150385571291870753516854020, −1.16540812991567514522518591861, 1.61115794801629939553685528769, 4.42888487255767125205107758312, 5.42812359295721825506069404812, 8.039889851515702274635933752128, 8.774279190993406096371375754687, 11.24467516944067464063560557108, 13.82175173093509564482094405044, 15.20548405326561858458673870934, 16.32264968719937597613789128470, 17.56221104980445347028991653407

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