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Properties

Label 2-3-3.2-c20-0-4
Degree $2$
Conductor $3$
Sign $-0.555 + 0.831i$
Analytic cond. $7.60541$
Root an. cond. $2.75779$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 552. i·2-s + (−3.27e4 + 4.91e4i)3-s + 7.43e5·4-s − 1.06e7i·5-s + (2.71e7 + 1.81e7i)6-s − 3.45e8·7-s − 9.90e8i·8-s + (−1.33e9 − 3.22e9i)9-s − 5.91e9·10-s − 3.65e10i·11-s + (−2.43e10 + 3.64e10i)12-s + 3.21e10·13-s + 1.91e11i·14-s + (5.25e11 + 3.50e11i)15-s + 2.31e11·16-s − 4.71e11i·17-s + ⋯
L(s)  = 1  − 0.539i·2-s + (−0.555 + 0.831i)3-s + 0.708·4-s − 1.09i·5-s + (0.448 + 0.299i)6-s − 1.22·7-s − 0.922i·8-s + (−0.383 − 0.923i)9-s − 0.591·10-s − 1.41i·11-s + (−0.393 + 0.589i)12-s + 0.233·13-s + 0.660i·14-s + (0.910 + 0.607i)15-s + 0.210·16-s − 0.234i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(7.60541\)
Root analytic conductor: \(2.75779\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :10),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.514727 - 0.962416i\)
\(L(\frac12)\) \(\approx\) \(0.514727 - 0.962416i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.27e4 - 4.91e4i)T \)
good2 \( 1 + 552. iT - 1.04e6T^{2} \)
5 \( 1 + 1.06e7iT - 9.53e13T^{2} \)
7 \( 1 + 3.45e8T + 7.97e16T^{2} \)
11 \( 1 + 3.65e10iT - 6.72e20T^{2} \)
13 \( 1 - 3.21e10T + 1.90e22T^{2} \)
17 \( 1 + 4.71e11iT - 4.06e24T^{2} \)
19 \( 1 + 3.83e12T + 3.75e25T^{2} \)
23 \( 1 - 5.00e13iT - 1.71e27T^{2} \)
29 \( 1 + 1.30e14iT - 1.76e29T^{2} \)
31 \( 1 + 1.26e15T + 6.71e29T^{2} \)
37 \( 1 - 5.32e15T + 2.31e31T^{2} \)
41 \( 1 + 8.65e15iT - 1.80e32T^{2} \)
43 \( 1 - 2.40e16T + 4.67e32T^{2} \)
47 \( 1 - 6.90e16iT - 2.76e33T^{2} \)
53 \( 1 + 1.83e17iT - 3.05e34T^{2} \)
59 \( 1 + 5.32e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.46e17T + 5.08e35T^{2} \)
67 \( 1 + 1.09e18T + 3.32e36T^{2} \)
71 \( 1 + 2.88e17iT - 1.05e37T^{2} \)
73 \( 1 - 4.12e18T + 1.84e37T^{2} \)
79 \( 1 - 1.16e19T + 8.96e37T^{2} \)
83 \( 1 + 1.45e19iT - 2.40e38T^{2} \)
89 \( 1 - 4.87e18iT - 9.72e38T^{2} \)
97 \( 1 + 3.51e19T + 5.43e39T^{2} \)
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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

−20.81488908225043982009936952080, −19.42944437907639235033147053382, −16.60852257963727818236846344341, −15.90809237115940260342212630318, −12.77224169520414038991758760710, −11.12975507351188891965806577298, −9.353893589905172118451877762938, −5.96053112742391075194481279316, −3.50613008580847392332752292687, −0.58646645850002603550907492205, 2.40109394107932512057203856572, 6.30517344065108310497058128710, 7.20437567994863498598857674468, 10.67794493310109250220587954808, 12.57681072040101325509698942527, 14.85322170729240247062732250119, 16.60072531953939955700186461798, 18.29681349466479722900734171887, 19.83437845907489822412372550890, 22.44848735990989167007265293369

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