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Properties

Label 2-3-3.2-c68-0-4
Degree $2$
Conductor $3$
Sign $-0.908 + 0.416i$
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  + 6.69e9i·2-s + (−1.51e16 + 6.95e15i)3-s + 2.50e20·4-s + 7.66e23i·5-s + (−4.65e25 − 1.01e26i)6-s + 4.76e28·7-s + 3.65e30i·8-s + (1.81e32 − 2.10e32i)9-s − 5.13e33·10-s − 1.05e35i·11-s + (−3.79e36 + 1.73e36i)12-s − 2.29e37·13-s + 3.18e38i·14-s + (−5.32e39 − 1.16e40i)15-s + 4.94e40·16-s + 9.58e41i·17-s + ⋯
L(s)  = 1  + 0.389i·2-s + (−0.908 + 0.416i)3-s + 0.848·4-s + 1.31i·5-s + (−0.162 − 0.354i)6-s + 0.880·7-s + 0.720i·8-s + (0.652 − 0.757i)9-s − 0.513·10-s − 0.411i·11-s + (−0.770 + 0.353i)12-s − 0.307·13-s + 0.343i·14-s + (−0.548 − 1.19i)15-s + 0.567·16-s + 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.908 + 0.416i$
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: \((2,\ 3,\ (\ :34),\ -0.908 + 0.416i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(1.447837772\)
\(L(\frac12)\) \(\approx\) \(1.447837772\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51e16 - 6.95e15i)T \)
good2 \( 1 - 6.69e9iT - 2.95e20T^{2} \)
5 \( 1 - 7.66e23iT - 3.38e47T^{2} \)
7 \( 1 - 4.76e28T + 2.92e57T^{2} \)
11 \( 1 + 1.05e35iT - 6.52e70T^{2} \)
13 \( 1 + 2.29e37T + 5.59e75T^{2} \)
17 \( 1 - 9.58e41iT - 4.68e83T^{2} \)
19 \( 1 + 4.30e43T + 9.02e86T^{2} \)
23 \( 1 + 2.39e46iT - 3.95e92T^{2} \)
29 \( 1 - 5.15e49iT - 2.77e99T^{2} \)
31 \( 1 - 8.13e50T + 2.58e101T^{2} \)
37 \( 1 + 2.30e52T + 4.34e106T^{2} \)
41 \( 1 - 2.46e54iT - 4.66e109T^{2} \)
43 \( 1 + 2.77e55T + 1.19e111T^{2} \)
47 \( 1 - 1.17e57iT - 5.04e113T^{2} \)
53 \( 1 - 5.27e58iT - 1.78e117T^{2} \)
59 \( 1 - 2.37e59iT - 2.61e120T^{2} \)
61 \( 1 + 2.42e60T + 2.52e121T^{2} \)
67 \( 1 + 5.51e61T + 1.48e124T^{2} \)
71 \( 1 + 1.62e63iT - 7.68e125T^{2} \)
73 \( 1 + 1.89e63T + 5.08e126T^{2} \)
79 \( 1 - 2.45e64T + 1.09e129T^{2} \)
83 \( 1 + 1.04e64iT - 3.14e130T^{2} \)
89 \( 1 - 3.23e65iT - 3.61e132T^{2} \)
97 \( 1 + 6.81e67T + 1.26e135T^{2} \)
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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

−14.61093311021186835279267361110, −12.21001555086239778851184708470, −10.80626986905250899472142385827, −10.64462289597038839871473432690, −8.153120539834080888815777946323, −6.68621367295026076134057174724, −6.09129055083826697195598953612, −4.49578062318650574360741351472, −2.88348910739724414247396786177, −1.54624278288403748218839656357, 0.34896329719228964313934021015, 1.33314464539118294941040254568, 2.20773801847767021698462894787, 4.43927427034591106675051767328, 5.37248652358382217456807785449, 6.86743396500157268965303466411, 8.122189244740623265511356878512, 9.958499702085684497406504422667, 11.46275045285565467524877568099, 12.08087271095095017444468342482

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