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Properties

Label 2-3-3.2-c68-0-16
Degree $2$
Conductor $3$
Sign $0.999 - 0.0374i$
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  + 2.79e10i·2-s + (1.66e16 − 6.25e14i)3-s − 4.85e20·4-s − 1.09e24i·5-s + (1.74e25 + 4.65e26i)6-s + 3.32e27·7-s − 5.31e30i·8-s + (2.77e32 − 2.08e31i)9-s + 3.05e34·10-s + 3.05e35i·11-s + (−8.08e36 + 3.03e35i)12-s + 7.37e37·13-s + 9.29e37i·14-s + (−6.82e38 − 1.82e40i)15-s + 5.15e39·16-s − 2.71e41i·17-s + ⋯
L(s)  = 1  + 1.62i·2-s + (0.999 − 0.0374i)3-s − 1.64·4-s − 1.87i·5-s + (0.0609 + 1.62i)6-s + 0.0614·7-s − 1.04i·8-s + (0.997 − 0.0749i)9-s + 3.05·10-s + 1.19i·11-s + (−1.64 + 0.0616i)12-s + 0.985·13-s + 0.0999i·14-s + (−0.0703 − 1.87i)15-s + 0.0591·16-s − 0.397i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0374i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (0.999 - 0.0374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.999 - 0.0374i$
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: \((2,\ 3,\ (\ :34),\ 0.999 - 0.0374i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(2.674654882\)
\(L(\frac12)\) \(\approx\) \(2.674654882\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66e16 + 6.25e14i)T \)
good2 \( 1 - 2.79e10iT - 2.95e20T^{2} \)
5 \( 1 + 1.09e24iT - 3.38e47T^{2} \)
7 \( 1 - 3.32e27T + 2.92e57T^{2} \)
11 \( 1 - 3.05e35iT - 6.52e70T^{2} \)
13 \( 1 - 7.37e37T + 5.59e75T^{2} \)
17 \( 1 + 2.71e41iT - 4.68e83T^{2} \)
19 \( 1 + 2.11e43T + 9.02e86T^{2} \)
23 \( 1 + 3.36e46iT - 3.95e92T^{2} \)
29 \( 1 + 1.39e48iT - 2.77e99T^{2} \)
31 \( 1 - 1.93e50T + 2.58e101T^{2} \)
37 \( 1 - 3.10e53T + 4.34e106T^{2} \)
41 \( 1 + 1.55e54iT - 4.66e109T^{2} \)
43 \( 1 + 2.69e55T + 1.19e111T^{2} \)
47 \( 1 + 2.47e56iT - 5.04e113T^{2} \)
53 \( 1 + 4.23e58iT - 1.78e117T^{2} \)
59 \( 1 - 1.31e59iT - 2.61e120T^{2} \)
61 \( 1 + 7.02e60T + 2.52e121T^{2} \)
67 \( 1 - 1.58e62T + 1.48e124T^{2} \)
71 \( 1 + 8.27e62iT - 7.68e125T^{2} \)
73 \( 1 - 2.43e63T + 5.08e126T^{2} \)
79 \( 1 - 3.21e64T + 1.09e129T^{2} \)
83 \( 1 + 1.90e65iT - 3.14e130T^{2} \)
89 \( 1 + 7.28e65iT - 3.61e132T^{2} \)
97 \( 1 - 1.57e67T + 1.26e135T^{2} \)
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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

−13.46510499803122759050513821785, −12.58939960742086980311290695622, −9.545407580829923412097679210101, −8.597035121933533836604567568614, −7.957828598644372265668293032112, −6.47620465758557231576971381462, −4.86476738033841556882682672059, −4.26209227948530360313919581062, −1.90698205680010947099572106087, −0.53915685751598268626260145940, 1.26728737431801341616149861266, 2.41116408603091397584322273303, 3.27426438244821264669443295717, 3.82287889520954310208004728832, 6.37336181656303621934792326459, 8.040642581923484467801472793268, 9.525316191290996889083631041866, 10.68710983811022388947548031596, 11.34543785596260284220790613266, 13.30200154024105339899996715124

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