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Properties

Label 2-3-3.2-c32-0-3
Degree $2$
Conductor $3$
Sign $-0.996 + 0.0822i$
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  + 5.92e4i·2-s + (−4.29e7 + 3.54e6i)3-s + 7.80e8·4-s + 2.23e11i·5-s + (−2.09e11 − 2.54e12i)6-s + 4.13e13·7-s + 3.00e14i·8-s + (1.82e15 − 3.03e14i)9-s − 1.32e16·10-s + 7.52e16i·11-s + (−3.35e16 + 2.76e15i)12-s + 3.61e17·13-s + 2.45e18i·14-s + (−7.91e17 − 9.58e18i)15-s − 1.44e19·16-s − 4.65e19i·17-s + ⋯
L(s)  = 1  + 0.904i·2-s + (−0.996 + 0.0822i)3-s + 0.181·4-s + 1.46i·5-s + (−0.0744 − 0.901i)6-s + 1.24·7-s + 1.06i·8-s + (0.986 − 0.163i)9-s − 1.32·10-s + 1.63i·11-s + (−0.181 + 0.0149i)12-s + 0.543·13-s + 1.12i·14-s + (−0.120 − 1.45i)15-s − 0.785·16-s − 0.956i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.996 + 0.0822i$
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: \((2,\ 3,\ (\ :16),\ -0.996 + 0.0822i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.0684853 - 1.66219i\)
\(L(\frac12)\) \(\approx\) \(0.0684853 - 1.66219i\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.29e7 - 3.54e6i)T \)
good2 \( 1 - 5.92e4iT - 4.29e9T^{2} \)
5 \( 1 - 2.23e11iT - 2.32e22T^{2} \)
7 \( 1 - 4.13e13T + 1.10e27T^{2} \)
11 \( 1 - 7.52e16iT - 2.11e33T^{2} \)
13 \( 1 - 3.61e17T + 4.42e35T^{2} \)
17 \( 1 + 4.65e19iT - 2.36e39T^{2} \)
19 \( 1 + 2.35e20T + 8.31e40T^{2} \)
23 \( 1 + 1.16e21iT - 3.76e43T^{2} \)
29 \( 1 - 6.23e22iT - 6.26e46T^{2} \)
31 \( 1 - 5.99e23T + 5.29e47T^{2} \)
37 \( 1 + 5.06e24T + 1.52e50T^{2} \)
41 \( 1 - 1.64e25iT - 4.06e51T^{2} \)
43 \( 1 - 7.52e25T + 1.86e52T^{2} \)
47 \( 1 + 3.86e26iT - 3.21e53T^{2} \)
53 \( 1 - 1.18e27iT - 1.50e55T^{2} \)
59 \( 1 + 2.66e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.94e28T + 1.35e57T^{2} \)
67 \( 1 - 3.59e28T + 2.71e58T^{2} \)
71 \( 1 + 7.06e29iT - 1.73e59T^{2} \)
73 \( 1 + 1.05e30T + 4.22e59T^{2} \)
79 \( 1 - 1.00e30T + 5.29e60T^{2} \)
83 \( 1 - 1.77e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.02e30iT - 2.40e62T^{2} \)
97 \( 1 - 8.20e31T + 3.77e63T^{2} \)
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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

−18.22136859243366379197572590363, −17.40272157107157146589658480758, −15.50560901934689565464147383161, −14.53174248553473069907627673098, −11.66231787402862068113374863596, −10.55931261224426839813496358353, −7.48248567057970773003047447621, −6.51615535718583523415997922064, −4.81827932898750655942509005667, −2.03084365240619974629838488881, 0.73280820389272097092063334738, 1.54102811666769356088107745088, 4.24031201318498397317921863752, 5.84109444520336281014100768578, 8.434708273025575105057429713512, 10.76556170631634776064153059510, 11.77568898512540366936259800376, 13.10566496796021517660626719276, 16.01873766115555060742368222038, 17.23258869784473774515271562241

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