Properties

Label 2-3-1.1-c67-0-4
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $85.2871$
Root an. cond. $9.23510$
Motivic weight $67$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.17e10·2-s − 5.55e15·3-s − 8.92e18·4-s + 8.66e22·5-s − 6.54e25·6-s + 3.38e28·7-s − 1.84e30·8-s + 3.09e31·9-s + 1.02e33·10-s − 2.84e34·11-s + 4.96e34·12-s + 5.77e36·13-s + 3.99e38·14-s − 4.81e38·15-s − 2.03e40·16-s − 1.58e41·17-s + 3.63e41·18-s + 1.30e43·19-s − 7.73e41·20-s − 1.88e44·21-s − 3.35e44·22-s + 5.95e44·23-s + 1.02e46·24-s − 6.02e46·25-s + 6.80e46·26-s − 1.71e47·27-s − 3.02e47·28-s + ⋯
L(s)  = 1  + 0.969·2-s − 0.577·3-s − 0.0605·4-s + 0.332·5-s − 0.559·6-s + 1.65·7-s − 1.02·8-s + 0.333·9-s + 0.322·10-s − 0.369·11-s + 0.0349·12-s + 0.278·13-s + 1.60·14-s − 0.192·15-s − 0.935·16-s − 0.957·17-s + 0.323·18-s + 1.89·19-s − 0.0201·20-s − 0.956·21-s − 0.358·22-s + 0.143·23-s + 0.593·24-s − 0.889·25-s + 0.269·26-s − 0.192·27-s − 0.100·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(85.2871\)
Root analytic conductor: \(9.23510\)
Motivic weight: \(67\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :67/2),\ 1)\)

Particular Values

\(L(34)\) \(\approx\) \(3.254019572\)
\(L(\frac12)\) \(\approx\) \(3.254019572\)
\(L(\frac{69}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.55e15T \)
good2 \( 1 - 1.17e10T + 1.47e20T^{2} \)
5 \( 1 - 8.66e22T + 6.77e46T^{2} \)
7 \( 1 - 3.38e28T + 4.18e56T^{2} \)
11 \( 1 + 2.84e34T + 5.93e69T^{2} \)
13 \( 1 - 5.77e36T + 4.30e74T^{2} \)
17 \( 1 + 1.58e41T + 2.75e82T^{2} \)
19 \( 1 - 1.30e43T + 4.74e85T^{2} \)
23 \( 1 - 5.95e44T + 1.72e91T^{2} \)
29 \( 1 + 1.88e48T + 9.56e97T^{2} \)
31 \( 1 + 9.76e49T + 8.34e99T^{2} \)
37 \( 1 + 2.00e52T + 1.17e105T^{2} \)
41 \( 1 - 1.76e54T + 1.13e108T^{2} \)
43 \( 1 - 5.65e54T + 2.76e109T^{2} \)
47 \( 1 - 1.13e56T + 1.07e112T^{2} \)
53 \( 1 - 5.57e57T + 3.36e115T^{2} \)
59 \( 1 - 2.54e59T + 4.43e118T^{2} \)
61 \( 1 + 4.65e59T + 4.14e119T^{2} \)
67 \( 1 - 7.13e59T + 2.22e122T^{2} \)
71 \( 1 - 1.41e62T + 1.08e124T^{2} \)
73 \( 1 + 2.86e62T + 6.96e124T^{2} \)
79 \( 1 - 2.81e63T + 1.38e127T^{2} \)
83 \( 1 - 3.57e64T + 3.78e128T^{2} \)
89 \( 1 - 2.86e64T + 4.06e130T^{2} \)
97 \( 1 - 4.69e66T + 1.29e133T^{2} \)
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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

−13.42986059716772123283000144034, −11.95519150438361005747514377227, −10.97544649159166527124277713177, −9.139860877135346242982796522328, −7.54450914171609948342417051246, −5.72967124991006947899200807819, −5.05585809684835227487684422942, −3.94894473150087184723454794389, −2.21130381419611809885876262625, −0.829137807603112330932751473845, 0.829137807603112330932751473845, 2.21130381419611809885876262625, 3.94894473150087184723454794389, 5.05585809684835227487684422942, 5.72967124991006947899200807819, 7.54450914171609948342417051246, 9.139860877135346242982796522328, 10.97544649159166527124277713177, 11.95519150438361005747514377227, 13.42986059716772123283000144034

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