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Properties

Label 2-3-3.2-c68-0-3
Degree $2$
Conductor $3$
Sign $0.593 + 0.804i$
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  − 3.11e10i·2-s + (9.90e15 + 1.34e16i)3-s − 6.72e20·4-s − 1.00e24i·5-s + (4.17e26 − 3.08e26i)6-s − 7.71e28·7-s + 1.17e31i·8-s + (−8.19e31 + 2.65e32i)9-s − 3.11e34·10-s + 3.39e34i·11-s + (−6.65e36 − 9.01e36i)12-s − 3.59e37·13-s + 2.40e39i·14-s + (1.34e40 − 9.90e39i)15-s + 1.66e41·16-s − 3.38e41i·17-s + ⋯
L(s)  = 1  − 1.81i·2-s + (0.593 + 0.804i)3-s − 2.27·4-s − 1.71i·5-s + (1.45 − 1.07i)6-s − 1.42·7-s + 2.31i·8-s + (−0.294 + 0.955i)9-s − 3.11·10-s + 0.133i·11-s + (−1.35 − 1.83i)12-s − 0.480·13-s + 2.58i·14-s + (1.38 − 1.02i)15-s + 1.90·16-s − 0.494i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.593 + 0.804i$
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: \((2,\ 3,\ (\ :34),\ 0.593 + 0.804i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(0.7718975252\)
\(L(\frac12)\) \(\approx\) \(0.7718975252\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.90e15 - 1.34e16i)T \)
good2 \( 1 + 3.11e10iT - 2.95e20T^{2} \)
5 \( 1 + 1.00e24iT - 3.38e47T^{2} \)
7 \( 1 + 7.71e28T + 2.92e57T^{2} \)
11 \( 1 - 3.39e34iT - 6.52e70T^{2} \)
13 \( 1 + 3.59e37T + 5.59e75T^{2} \)
17 \( 1 + 3.38e41iT - 4.68e83T^{2} \)
19 \( 1 + 5.52e42T + 9.02e86T^{2} \)
23 \( 1 - 2.18e46iT - 3.95e92T^{2} \)
29 \( 1 + 6.20e49iT - 2.77e99T^{2} \)
31 \( 1 - 3.37e50T + 2.58e101T^{2} \)
37 \( 1 + 1.68e53T + 4.34e106T^{2} \)
41 \( 1 - 1.19e55iT - 4.66e109T^{2} \)
43 \( 1 - 5.90e54T + 1.19e111T^{2} \)
47 \( 1 + 8.74e56iT - 5.04e113T^{2} \)
53 \( 1 - 3.44e57iT - 1.78e117T^{2} \)
59 \( 1 + 2.11e60iT - 2.61e120T^{2} \)
61 \( 1 - 8.52e60T + 2.52e121T^{2} \)
67 \( 1 - 1.56e61T + 1.48e124T^{2} \)
71 \( 1 + 3.00e62iT - 7.68e125T^{2} \)
73 \( 1 - 1.68e63T + 5.08e126T^{2} \)
79 \( 1 + 5.19e64T + 1.09e129T^{2} \)
83 \( 1 - 1.50e65iT - 3.14e130T^{2} \)
89 \( 1 - 8.80e65iT - 3.61e132T^{2} \)
97 \( 1 - 7.23e66T + 1.26e135T^{2} \)
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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

−12.87186373853867783425041949250, −11.71604417308044844513320183758, −9.823741496857571492456378201507, −9.553721215145846708231024063729, −8.397792681360977948600517999379, −5.22219421050444725430830659566, −4.23859326837052774378431869130, −3.25910990295399671365334759474, −2.07415064611882247230151985884, −0.70657352822655065162282929226, 0.23806840384172927829641780463, 2.63860727808743634510451851122, 3.69978314465152297552051364975, 6.00136283118198411838414713741, 6.73841373734992700092430126471, 7.31023578015689800038316240204, 8.760335444246610643490124946006, 10.15269597190335885877784793730, 12.70887256106784723278066836192, 13.98762093252074105276904172046

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