Properties

Label 2-3-3.2-c68-0-10
Degree $2$
Conductor $3$
Sign $-0.787 - 0.616i$
Analytic cond. $87.8517$
Root an. cond. $9.37292$
Motivic weight $68$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.12e10i·2-s + (−1.31e16 − 1.02e16i)3-s − 6.79e20·4-s + 1.89e23i·5-s + (−3.20e26 + 4.09e26i)6-s + 1.94e27·7-s + 1.19e31i·8-s + (6.68e31 + 2.69e32i)9-s + 5.91e33·10-s + 3.64e35i·11-s + (8.92e36 + 6.98e36i)12-s + 2.03e37·13-s − 6.05e37i·14-s + (1.94e39 − 2.48e39i)15-s + 1.73e41·16-s + 9.09e41i·17-s + ⋯
L(s)  = 1  − 1.81i·2-s + (−0.787 − 0.616i)3-s − 2.30·4-s + 0.325i·5-s + (−1.11 + 1.43i)6-s + 0.0358·7-s + 2.36i·8-s + (0.240 + 0.970i)9-s + 0.591·10-s + 1.42i·11-s + (1.81 + 1.41i)12-s + 0.271·13-s − 0.0651i·14-s + (0.200 − 0.256i)15-s + 1.99·16-s + 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(87.8517\)
Root analytic conductor: \(9.37292\)
Motivic weight: \(68\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :34),\ -0.787 - 0.616i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(0.4431192378\)
\(L(\frac12)\) \(\approx\) \(0.4431192378\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.31e16 + 1.02e16i)T \)
good2 \( 1 + 3.12e10iT - 2.95e20T^{2} \)
5 \( 1 - 1.89e23iT - 3.38e47T^{2} \)
7 \( 1 - 1.94e27T + 2.92e57T^{2} \)
11 \( 1 - 3.64e35iT - 6.52e70T^{2} \)
13 \( 1 - 2.03e37T + 5.59e75T^{2} \)
17 \( 1 - 9.09e41iT - 4.68e83T^{2} \)
19 \( 1 + 4.81e43T + 9.02e86T^{2} \)
23 \( 1 + 1.99e46iT - 3.95e92T^{2} \)
29 \( 1 - 5.94e49iT - 2.77e99T^{2} \)
31 \( 1 + 8.63e50T + 2.58e101T^{2} \)
37 \( 1 + 2.63e53T + 4.34e106T^{2} \)
41 \( 1 + 4.92e54iT - 4.66e109T^{2} \)
43 \( 1 - 3.71e54T + 1.19e111T^{2} \)
47 \( 1 + 6.88e56iT - 5.04e113T^{2} \)
53 \( 1 - 3.74e58iT - 1.78e117T^{2} \)
59 \( 1 + 7.67e59iT - 2.61e120T^{2} \)
61 \( 1 - 5.35e60T + 2.52e121T^{2} \)
67 \( 1 - 2.01e62T + 1.48e124T^{2} \)
71 \( 1 + 5.82e62iT - 7.68e125T^{2} \)
73 \( 1 - 8.80e62T + 5.08e126T^{2} \)
79 \( 1 + 1.85e63T + 1.09e129T^{2} \)
83 \( 1 + 9.38e64iT - 3.14e130T^{2} \)
89 \( 1 - 8.09e65iT - 3.61e132T^{2} \)
97 \( 1 - 2.56e67T + 1.26e135T^{2} \)
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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

−12.52869360096584274689227225924, −10.89502475273237288202182182006, −10.38328419048249887882153618205, −8.629690155189102973159202982406, −6.77320449362799715259824549906, −5.00139554048125668961012517700, −3.85948914711009198412799685835, −2.16758616561995404856247987425, −1.62422316641279404053542171816, −0.17694319177395734365720127248, 0.68715978199529381687752575227, 3.70675125720611031934281566655, 4.95218153000278866976657390556, 5.79786255485321773386252202715, 6.82376400049483530844637470681, 8.393521535078915710387214932401, 9.399140569042808281343733837053, 11.17316664496499330277988794017, 13.10146460514735564018241971817, 14.45795684516991476345792544245

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