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Properties

Label 2-3-3.2-c32-0-9
Degree $2$
Conductor $3$
Sign $0.565 - 0.824i$
Analytic cond. $19.4599$
Root an. cond. $4.41134$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.26e5i·2-s + (2.43e7 − 3.54e7i)3-s − 1.18e10·4-s − 1.55e11i·5-s + (−4.50e12 − 3.08e12i)6-s + 3.32e12·7-s + 9.52e14i·8-s + (−6.66e14 − 1.72e15i)9-s − 1.97e16·10-s + 8.04e15i·11-s + (−2.87e17 + 4.19e17i)12-s − 2.41e17·13-s − 4.21e17i·14-s + (−5.52e18 − 3.79e18i)15-s + 7.02e19·16-s − 3.34e19i·17-s + ⋯
L(s)  = 1  − 1.93i·2-s + (0.565 − 0.824i)3-s − 2.74·4-s − 1.02i·5-s + (−1.59 − 1.09i)6-s + 0.100·7-s + 3.38i·8-s + (−0.359 − 0.932i)9-s − 1.97·10-s + 0.175i·11-s + (−1.55 + 2.26i)12-s − 0.363·13-s − 0.193i·14-s + (−0.841 − 0.577i)15-s + 3.80·16-s − 0.687i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(19.4599\)
Root analytic conductor: \(4.41134\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :16),\ 0.565 - 0.824i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.14082 + 0.600823i\)
\(L(\frac12)\) \(\approx\) \(1.14082 + 0.600823i\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.43e7 + 3.54e7i)T \)
good2 \( 1 + 1.26e5iT - 4.29e9T^{2} \)
5 \( 1 + 1.55e11iT - 2.32e22T^{2} \)
7 \( 1 - 3.32e12T + 1.10e27T^{2} \)
11 \( 1 - 8.04e15iT - 2.11e33T^{2} \)
13 \( 1 + 2.41e17T + 4.42e35T^{2} \)
17 \( 1 + 3.34e19iT - 2.36e39T^{2} \)
19 \( 1 - 1.91e20T + 8.31e40T^{2} \)
23 \( 1 + 4.15e21iT - 3.76e43T^{2} \)
29 \( 1 + 1.41e23iT - 6.26e46T^{2} \)
31 \( 1 - 9.52e23T + 5.29e47T^{2} \)
37 \( 1 + 2.23e25T + 1.52e50T^{2} \)
41 \( 1 - 7.36e25iT - 4.06e51T^{2} \)
43 \( 1 + 9.07e25T + 1.86e52T^{2} \)
47 \( 1 - 7.60e26iT - 3.21e53T^{2} \)
53 \( 1 - 1.49e27iT - 1.50e55T^{2} \)
59 \( 1 + 1.18e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.17e28T + 1.35e57T^{2} \)
67 \( 1 - 2.96e29T + 2.71e58T^{2} \)
71 \( 1 + 5.78e29iT - 1.73e59T^{2} \)
73 \( 1 + 6.43e29T + 4.22e59T^{2} \)
79 \( 1 + 1.65e30T + 5.29e60T^{2} \)
83 \( 1 + 1.50e29iT - 2.57e61T^{2} \)
89 \( 1 + 1.77e31iT - 2.40e62T^{2} \)
97 \( 1 - 1.01e31T + 3.77e63T^{2} \)
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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

−17.55995082846107783957266327163, −13.99692346526450216952588515775, −12.78337330429612628133948707781, −11.79735929664806947012760200009, −9.620254307090323008787532436863, −8.357316252107375831249992540543, −4.75048784892673222655195664649, −2.94573668541317151540832919514, −1.54345496345849636242513314267, −0.47066966545282972853746542975, 3.56923892365925808228323183050, 5.26196543948011987255472916091, 7.00397506671824918219995769820, 8.463849808362742808909610119519, 10.06733978348014810179071599907, 13.81432209978175735694597787482, 14.80935025739543256792625306984, 15.83429309893729128044354363620, 17.37379618929060630876438842838, 18.99959824826754534034480716966

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