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Properties

Label 2-3-3.2-c32-0-0
Degree $2$
Conductor $3$
Sign $-0.152 - 0.988i$
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  − 3.25e4i·2-s + (−6.54e6 − 4.25e7i)3-s + 3.23e9·4-s + 1.19e11i·5-s + (−1.38e12 + 2.13e11i)6-s − 3.44e13·7-s − 2.45e14i·8-s + (−1.76e15 + 5.56e14i)9-s + 3.88e15·10-s + 2.29e16i·11-s + (−2.11e16 − 1.37e17i)12-s − 7.69e17·13-s + 1.12e18i·14-s + (5.07e18 − 7.81e17i)15-s + 5.89e18·16-s + 2.61e19i·17-s + ⋯
L(s)  = 1  − 0.497i·2-s + (−0.152 − 0.988i)3-s + 0.752·4-s + 0.782i·5-s + (−0.491 + 0.0755i)6-s − 1.03·7-s − 0.871i·8-s + (−0.953 + 0.300i)9-s + 0.388·10-s + 0.498i·11-s + (−0.114 − 0.744i)12-s − 1.15·13-s + 0.515i·14-s + (0.772 − 0.118i)15-s + 0.319·16-s + 0.537i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.152 - 0.988i$
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: \((2,\ 3,\ (\ :16),\ -0.152 - 0.988i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.204203 + 0.238020i\)
\(L(\frac12)\) \(\approx\) \(0.204203 + 0.238020i\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.54e6 + 4.25e7i)T \)
good2 \( 1 + 3.25e4iT - 4.29e9T^{2} \)
5 \( 1 - 1.19e11iT - 2.32e22T^{2} \)
7 \( 1 + 3.44e13T + 1.10e27T^{2} \)
11 \( 1 - 2.29e16iT - 2.11e33T^{2} \)
13 \( 1 + 7.69e17T + 4.42e35T^{2} \)
17 \( 1 - 2.61e19iT - 2.36e39T^{2} \)
19 \( 1 + 3.34e20T + 8.31e40T^{2} \)
23 \( 1 - 1.19e22iT - 3.76e43T^{2} \)
29 \( 1 + 2.90e23iT - 6.26e46T^{2} \)
31 \( 1 - 3.11e23T + 5.29e47T^{2} \)
37 \( 1 + 5.78e24T + 1.52e50T^{2} \)
41 \( 1 - 3.53e25iT - 4.06e51T^{2} \)
43 \( 1 + 1.98e26T + 1.86e52T^{2} \)
47 \( 1 + 5.91e26iT - 3.21e53T^{2} \)
53 \( 1 - 3.37e27iT - 1.50e55T^{2} \)
59 \( 1 + 2.27e28iT - 4.64e56T^{2} \)
61 \( 1 + 1.22e28T + 1.35e57T^{2} \)
67 \( 1 + 2.88e29T + 2.71e58T^{2} \)
71 \( 1 - 5.23e29iT - 1.73e59T^{2} \)
73 \( 1 + 2.42e29T + 4.22e59T^{2} \)
79 \( 1 + 6.06e29T + 5.29e60T^{2} \)
83 \( 1 + 9.99e30iT - 2.57e61T^{2} \)
89 \( 1 - 6.79e30iT - 2.40e62T^{2} \)
97 \( 1 - 8.12e31T + 3.77e63T^{2} \)
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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

−19.03926633327464779540993516014, −17.14141834676348548872955982264, −15.11388468551184002180786685453, −13.01798961489666210171554436485, −11.75644962164778363998835644232, −10.11717202738144457321258529103, −7.32502971354830161755032128168, −6.31027480787758493730998507856, −3.08279953417597026079766488255, −1.89926867162753171766414309590, 0.10339536107275742927649452116, 2.82213356571156191330043390629, 4.88236418382632927017132758345, 6.48854823824366459547250315720, 8.745779856765296463361874522672, 10.45547739838041969908048229322, 12.32625308513304313856504808537, 14.75518323408600181618475456716, 16.24651571135173498588585486619, 16.80878927333532584335129659234

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