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Properties

Label 2-3-3.2-c22-0-2
Degree $2$
Conductor $3$
Sign $-0.272 - 0.962i$
Analytic cond. $9.20122$
Root an. cond. $3.03335$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 634. i·2-s + (4.82e4 + 1.70e5i)3-s + 3.79e6·4-s + 2.56e6i·5-s + (−1.08e8 + 3.06e7i)6-s + 8.97e8·7-s + 5.06e9i·8-s + (−2.67e10 + 1.64e10i)9-s − 1.62e9·10-s + 4.54e11i·11-s + (1.82e11 + 6.46e11i)12-s − 1.52e12·13-s + 5.69e11i·14-s + (−4.37e11 + 1.23e11i)15-s + 1.26e13·16-s − 5.06e13i·17-s + ⋯
L(s)  = 1  + 0.309i·2-s + (0.272 + 0.962i)3-s + 0.903·4-s + 0.0525i·5-s + (−0.298 + 0.0843i)6-s + 0.453·7-s + 0.589i·8-s + (−0.851 + 0.523i)9-s − 0.0162·10-s + 1.59i·11-s + (0.246 + 0.869i)12-s − 0.853·13-s + 0.140i·14-s + (−0.0505 + 0.0143i)15-s + 0.721·16-s − 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(9.20122\)
Root analytic conductor: \(3.03335\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :11),\ -0.272 - 0.962i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(1.32587 + 1.75308i\)
\(L(\frac12)\) \(\approx\) \(1.32587 + 1.75308i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.82e4 - 1.70e5i)T \)
good2 \( 1 - 634. iT - 4.19e6T^{2} \)
5 \( 1 - 2.56e6iT - 2.38e15T^{2} \)
7 \( 1 - 8.97e8T + 3.90e18T^{2} \)
11 \( 1 - 4.54e11iT - 8.14e22T^{2} \)
13 \( 1 + 1.52e12T + 3.21e24T^{2} \)
17 \( 1 + 5.06e13iT - 1.17e27T^{2} \)
19 \( 1 - 2.10e13T + 1.35e28T^{2} \)
23 \( 1 - 1.03e15iT - 9.07e29T^{2} \)
29 \( 1 + 1.09e16iT - 1.48e32T^{2} \)
31 \( 1 - 2.32e16T + 6.45e32T^{2} \)
37 \( 1 - 2.29e17T + 3.16e34T^{2} \)
41 \( 1 + 7.03e17iT - 3.02e35T^{2} \)
43 \( 1 - 3.20e17T + 8.63e35T^{2} \)
47 \( 1 + 3.71e17iT - 6.11e36T^{2} \)
53 \( 1 + 1.31e19iT - 8.59e37T^{2} \)
59 \( 1 - 2.62e19iT - 9.09e38T^{2} \)
61 \( 1 + 3.33e19T + 1.89e39T^{2} \)
67 \( 1 - 1.84e20T + 1.49e40T^{2} \)
71 \( 1 - 6.16e19iT - 5.34e40T^{2} \)
73 \( 1 - 5.98e19T + 9.84e40T^{2} \)
79 \( 1 + 1.05e21T + 5.59e41T^{2} \)
83 \( 1 + 1.35e21iT - 1.65e42T^{2} \)
89 \( 1 + 7.46e20iT - 7.70e42T^{2} \)
97 \( 1 - 5.46e21T + 5.11e43T^{2} \)
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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

−20.96098431123844988269423725674, −19.99065625101592616180813789725, −17.31389309650569333828971308908, −15.70640888623659606512632076639, −14.58993699329112099289011940502, −11.65128453302186277118936977415, −9.821209243303868121360767550388, −7.43624434275993137945998008811, −4.92285971224108343039870424030, −2.46204756692539834131464786000, 1.11857598096344048042108222658, 2.82755296976513199969678474019, 6.31652993605159054143179566966, 8.168967554166307530684594486665, 11.05443217086022982380889701209, 12.61186867552150480186153977743, 14.57781254543108331711248349463, 16.78855229930212350669959814707, 18.81886103618296833602616904516, 19.99961275844871185341235799357

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