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Properties

Label 2-3-3.2-c68-0-21
Degree $2$
Conductor $3$
Sign $0.415 - 0.909i$
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  − 2.34e10i·2-s + (6.92e15 − 1.51e16i)3-s − 2.56e20·4-s − 4.36e23i·5-s + (−3.56e26 − 1.62e26i)6-s + 1.55e28·7-s − 9.07e29i·8-s + (−1.82e32 − 2.10e32i)9-s − 1.02e34·10-s − 1.73e35i·11-s + (−1.77e36 + 3.89e36i)12-s + 1.47e37·13-s − 3.65e38i·14-s + (−6.62e39 − 3.02e39i)15-s − 9.70e40·16-s − 9.48e41i·17-s + ⋯
L(s)  = 1  − 1.36i·2-s + (0.415 − 0.909i)3-s − 0.869·4-s − 0.749i·5-s + (−1.24 − 0.567i)6-s + 0.287·7-s − 0.178i·8-s + (−0.655 − 0.755i)9-s − 1.02·10-s − 0.677i·11-s + (−0.360 + 0.790i)12-s + 0.197·13-s − 0.393i·14-s + (−0.682 − 0.311i)15-s − 1.11·16-s − 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.415 - 0.909i$
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: \((2,\ 3,\ (\ :34),\ 0.415 - 0.909i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(2.217601484\)
\(L(\frac12)\) \(\approx\) \(2.217601484\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.92e15 + 1.51e16i)T \)
good2 \( 1 + 2.34e10iT - 2.95e20T^{2} \)
5 \( 1 + 4.36e23iT - 3.38e47T^{2} \)
7 \( 1 - 1.55e28T + 2.92e57T^{2} \)
11 \( 1 + 1.73e35iT - 6.52e70T^{2} \)
13 \( 1 - 1.47e37T + 5.59e75T^{2} \)
17 \( 1 + 9.48e41iT - 4.68e83T^{2} \)
19 \( 1 - 3.50e42T + 9.02e86T^{2} \)
23 \( 1 + 3.45e46iT - 3.95e92T^{2} \)
29 \( 1 - 3.52e49iT - 2.77e99T^{2} \)
31 \( 1 - 2.41e50T + 2.58e101T^{2} \)
37 \( 1 - 7.91e52T + 4.34e106T^{2} \)
41 \( 1 + 1.55e54iT - 4.66e109T^{2} \)
43 \( 1 + 2.47e55T + 1.19e111T^{2} \)
47 \( 1 - 1.05e57iT - 5.04e113T^{2} \)
53 \( 1 + 2.61e55iT - 1.78e117T^{2} \)
59 \( 1 + 1.97e60iT - 2.61e120T^{2} \)
61 \( 1 - 6.88e60T + 2.52e121T^{2} \)
67 \( 1 + 1.55e62T + 1.48e124T^{2} \)
71 \( 1 - 6.03e62iT - 7.68e125T^{2} \)
73 \( 1 - 4.34e63T + 5.08e126T^{2} \)
79 \( 1 - 5.77e64T + 1.09e129T^{2} \)
83 \( 1 + 8.18e64iT - 3.14e130T^{2} \)
89 \( 1 - 2.02e66iT - 3.61e132T^{2} \)
97 \( 1 - 6.55e66T + 1.26e135T^{2} \)
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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

−12.13465665368393106428924657851, −11.04916372149421002319468707792, −9.344927043507570797935972095976, −8.304050664317173988262049671230, −6.64689156134002562047955435337, −4.80404939176890474712992753649, −3.24544044164543849931692579674, −2.30042345416297235168180903266, −1.05632144958231325452497399127, −0.51671770778416353552445226115, 2.04921539126583569238472359184, 3.58180912389758124150295295991, 4.89586440852520311864208470808, 6.10381345057278228450611340048, 7.45252189224883526577041213252, 8.500205048451113656228080260009, 9.953390716367506948030344992107, 11.28596076864247018675957509409, 13.66157954641538173598455417711, 14.91563592228361588454121624453

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