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Properties

Label 2-3-3.2-c68-0-0
Degree $2$
Conductor $3$
Sign $-0.382 - 0.924i$
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  − 9.08e9i·2-s + (−6.37e15 − 1.54e16i)3-s + 2.12e20·4-s + 8.19e23i·5-s + (−1.39e26 + 5.78e25i)6-s − 1.45e28·7-s − 4.61e30i·8-s + (−1.96e32 + 1.96e32i)9-s + 7.44e33·10-s − 9.57e34i·11-s + (−1.35e36 − 3.27e36i)12-s − 4.01e37·13-s + 1.32e38i·14-s + (1.26e40 − 5.22e39i)15-s + 2.08e40·16-s + 9.03e40i·17-s + ⋯
L(s)  = 1  − 0.528i·2-s + (−0.382 − 0.924i)3-s + 0.720·4-s + 1.40i·5-s + (−0.488 + 0.202i)6-s − 0.269·7-s − 0.909i·8-s + (−0.707 + 0.706i)9-s + 0.744·10-s − 0.374i·11-s + (−0.275 − 0.665i)12-s − 0.536·13-s + 0.142i·14-s + (1.30 − 0.538i)15-s + 0.239·16-s + 0.132i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.382 - 0.924i$
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: \((2,\ 3,\ (\ :34),\ -0.382 - 0.924i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(0.2804914845\)
\(L(\frac12)\) \(\approx\) \(0.2804914845\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.37e15 + 1.54e16i)T \)
good2 \( 1 + 9.08e9iT - 2.95e20T^{2} \)
5 \( 1 - 8.19e23iT - 3.38e47T^{2} \)
7 \( 1 + 1.45e28T + 2.92e57T^{2} \)
11 \( 1 + 9.57e34iT - 6.52e70T^{2} \)
13 \( 1 + 4.01e37T + 5.59e75T^{2} \)
17 \( 1 - 9.03e40iT - 4.68e83T^{2} \)
19 \( 1 - 5.00e43T + 9.02e86T^{2} \)
23 \( 1 - 1.67e45iT - 3.95e92T^{2} \)
29 \( 1 + 6.61e49iT - 2.77e99T^{2} \)
31 \( 1 + 7.76e50T + 2.58e101T^{2} \)
37 \( 1 + 5.29e52T + 4.34e106T^{2} \)
41 \( 1 - 7.25e54iT - 4.66e109T^{2} \)
43 \( 1 + 4.70e55T + 1.19e111T^{2} \)
47 \( 1 + 3.33e56iT - 5.04e113T^{2} \)
53 \( 1 - 2.18e58iT - 1.78e117T^{2} \)
59 \( 1 - 2.70e60iT - 2.61e120T^{2} \)
61 \( 1 - 5.80e60T + 2.52e121T^{2} \)
67 \( 1 + 1.64e62T + 1.48e124T^{2} \)
71 \( 1 + 8.24e62iT - 7.68e125T^{2} \)
73 \( 1 + 1.92e63T + 5.08e126T^{2} \)
79 \( 1 + 6.09e64T + 1.09e129T^{2} \)
83 \( 1 - 2.89e65iT - 3.14e130T^{2} \)
89 \( 1 + 1.30e65iT - 3.61e132T^{2} \)
97 \( 1 + 4.75e66T + 1.26e135T^{2} \)
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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

−13.43161232226311144073633749544, −11.91735148723000539265966065501, −11.16432508899050295022990562595, −9.986693980564217542110243347252, −7.59045692962650438658496157546, −6.84226366773787246223513042673, −5.77257881349168824161990849879, −3.30519693882398713767441799612, −2.55111965790338455444550809930, −1.36988445925166146611744891955, 0.06151206797155422349757631976, 1.52146166128221062406800877507, 3.28665932767416200215022984714, 4.91323488021046375476619312031, 5.53959224992974813086061998824, 7.17784321242697899298332805210, 8.729208381963463219249836270908, 9.871305472017764987877388501491, 11.46371678827636721176379124807, 12.53727190461241590750769674664

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