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Properties

Label 2-3-3.2-c68-0-5
Degree $2$
Conductor $3$
Sign $-0.913 - 0.407i$
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  + 1.98e10i·2-s + (−1.52e16 − 6.79e15i)3-s − 9.94e19·4-s + 1.15e23i·5-s + (1.34e26 − 3.02e26i)6-s − 6.02e28·7-s + 3.88e30i·8-s + (1.85e32 + 2.06e32i)9-s − 2.28e33·10-s + 2.49e35i·11-s + (1.51e36 + 6.75e35i)12-s + 6.80e37·13-s − 1.19e39i·14-s + (7.82e38 − 1.75e39i)15-s − 1.06e41·16-s − 5.64e41i·17-s + ⋯
L(s)  = 1  + 1.15i·2-s + (−0.913 − 0.407i)3-s − 0.336·4-s + 0.197i·5-s + (0.471 − 1.05i)6-s − 1.11·7-s + 0.766i·8-s + (0.667 + 0.744i)9-s − 0.228·10-s + 0.974i·11-s + (0.307 + 0.137i)12-s + 0.909·13-s − 1.28i·14-s + (0.0806 − 0.180i)15-s − 1.22·16-s − 0.824i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.913 - 0.407i$
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: \((2,\ 3,\ (\ :34),\ -0.913 - 0.407i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(1.396470735\)
\(L(\frac12)\) \(\approx\) \(1.396470735\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52e16 + 6.79e15i)T \)
good2 \( 1 - 1.98e10iT - 2.95e20T^{2} \)
5 \( 1 - 1.15e23iT - 3.38e47T^{2} \)
7 \( 1 + 6.02e28T + 2.92e57T^{2} \)
11 \( 1 - 2.49e35iT - 6.52e70T^{2} \)
13 \( 1 - 6.80e37T + 5.59e75T^{2} \)
17 \( 1 + 5.64e41iT - 4.68e83T^{2} \)
19 \( 1 - 3.74e43T + 9.02e86T^{2} \)
23 \( 1 + 9.44e45iT - 3.95e92T^{2} \)
29 \( 1 + 4.04e49iT - 2.77e99T^{2} \)
31 \( 1 - 3.19e50T + 2.58e101T^{2} \)
37 \( 1 - 5.63e52T + 4.34e106T^{2} \)
41 \( 1 - 5.48e54iT - 4.66e109T^{2} \)
43 \( 1 - 5.73e55T + 1.19e111T^{2} \)
47 \( 1 - 1.10e57iT - 5.04e113T^{2} \)
53 \( 1 - 5.86e58iT - 1.78e117T^{2} \)
59 \( 1 + 2.64e60iT - 2.61e120T^{2} \)
61 \( 1 + 3.65e60T + 2.52e121T^{2} \)
67 \( 1 - 1.10e62T + 1.48e124T^{2} \)
71 \( 1 - 2.24e62iT - 7.68e125T^{2} \)
73 \( 1 - 6.26e62T + 5.08e126T^{2} \)
79 \( 1 + 3.84e63T + 1.09e129T^{2} \)
83 \( 1 - 1.42e65iT - 3.14e130T^{2} \)
89 \( 1 + 1.80e66iT - 3.61e132T^{2} \)
97 \( 1 + 5.21e67T + 1.26e135T^{2} \)
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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

−13.89061348819576739393709119776, −12.51827096848482110424121608427, −11.13811024473392962797413852430, −9.570380394881818864739655486599, −7.67648694056100182187897201886, −6.73586273699329695595733189285, −5.96536975305437429667073158105, −4.67476841734027241773827666373, −2.67997439741773180716313880175, −0.948348041057826106542975378015, 0.48270880092853324030026562607, 1.24946159486557777721363679616, 3.12151682642953859234820586899, 3.86029347955727144758460791281, 5.64306135971208802329015668603, 6.73386789049288807499445454645, 9.079550051711562400825853323111, 10.26117894718831894472438195825, 11.17109635908136999836432309523, 12.30829076134889927422289576026

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