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Properties

Label 2-3-3.2-c18-0-0
Degree $2$
Conductor $3$
Sign $0.215 - 0.976i$
Analytic cond. $6.16158$
Root an. cond. $2.48225$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 932. i·2-s + (−4.23e3 + 1.92e4i)3-s − 6.07e5·4-s + 1.14e6i·5-s + (1.79e7 + 3.94e6i)6-s − 9.81e6·7-s + 3.22e8i·8-s + (−3.51e8 − 1.62e8i)9-s + 1.06e9·10-s + 2.16e9i·11-s + (2.57e9 − 1.16e10i)12-s − 1.47e10·13-s + 9.15e9i·14-s + (−2.19e10 − 4.83e9i)15-s + 1.41e11·16-s + 1.36e10i·17-s + ⋯
L(s)  = 1  − 1.82i·2-s + (−0.215 + 0.976i)3-s − 2.31·4-s + 0.584i·5-s + (1.77 + 0.391i)6-s − 0.243·7-s + 2.40i·8-s + (−0.907 − 0.420i)9-s + 1.06·10-s + 0.917i·11-s + (0.498 − 2.26i)12-s − 1.38·13-s + 0.442i·14-s + (−0.571 − 0.125i)15-s + 2.05·16-s + 0.115i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(6.16158\)
Root analytic conductor: \(2.48225\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :9),\ 0.215 - 0.976i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.288210 + 0.231636i\)
\(L(\frac12)\) \(\approx\) \(0.288210 + 0.231636i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.23e3 - 1.92e4i)T \)
good2 \( 1 + 932. iT - 2.62e5T^{2} \)
5 \( 1 - 1.14e6iT - 3.81e12T^{2} \)
7 \( 1 + 9.81e6T + 1.62e15T^{2} \)
11 \( 1 - 2.16e9iT - 5.55e18T^{2} \)
13 \( 1 + 1.47e10T + 1.12e20T^{2} \)
17 \( 1 - 1.36e10iT - 1.40e22T^{2} \)
19 \( 1 + 2.38e11T + 1.04e23T^{2} \)
23 \( 1 + 5.70e11iT - 3.24e24T^{2} \)
29 \( 1 + 1.35e13iT - 2.10e26T^{2} \)
31 \( 1 + 1.17e13T + 6.99e26T^{2} \)
37 \( 1 + 1.26e14T + 1.68e28T^{2} \)
41 \( 1 - 3.28e14iT - 1.07e29T^{2} \)
43 \( 1 - 8.40e14T + 2.52e29T^{2} \)
47 \( 1 - 9.97e14iT - 1.25e30T^{2} \)
53 \( 1 - 5.45e15iT - 1.08e31T^{2} \)
59 \( 1 + 8.03e14iT - 7.50e31T^{2} \)
61 \( 1 - 1.04e16T + 1.36e32T^{2} \)
67 \( 1 + 1.50e16T + 7.40e32T^{2} \)
71 \( 1 + 4.47e16iT - 2.10e33T^{2} \)
73 \( 1 + 3.92e16T + 3.46e33T^{2} \)
79 \( 1 + 7.20e16T + 1.43e34T^{2} \)
83 \( 1 + 1.02e17iT - 3.49e34T^{2} \)
89 \( 1 - 5.12e17iT - 1.22e35T^{2} \)
97 \( 1 + 7.28e17T + 5.77e35T^{2} \)
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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

−21.99866719651942617296410302882, −20.66093401767692633877397390720, −19.32934560951804847129823707393, −17.47922127685418936653693802840, −14.66553677059359858556372116360, −12.33252076187058749702464154719, −10.69116032561887396356713062181, −9.549559944850293134810634436874, −4.47006469852609182577849627871, −2.64563251124459065798211352237, 0.21279680117724765709893767246, 5.36882103519765579444020504273, 7.04734512498114625163189500203, 8.654058992399360153589039282056, 12.80916506798079875930602272471, 14.35510480299154864627661117601, 16.39736593526527573529566357611, 17.47352518751496202128310732232, 19.14884204022160949633830844544, 22.36990056644114839035632200611

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