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Properties

Label 2-3-3.2-c68-0-6
Degree $2$
Conductor $3$
Sign $-0.341 - 0.939i$
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  + 2.53e10i·2-s + (−5.69e15 − 1.56e16i)3-s − 3.49e20·4-s − 2.74e23i·5-s + (3.98e26 − 1.44e26i)6-s + 9.96e28·7-s − 1.37e30i·8-s + (−2.13e32 + 1.78e32i)9-s + 6.97e33·10-s − 1.53e35i·11-s + (1.98e36 + 5.47e36i)12-s − 1.32e38·13-s + 2.53e39i·14-s + (−4.30e39 + 1.56e39i)15-s − 6.81e40·16-s + 5.44e41i·17-s + ⋯
L(s)  = 1  + 1.47i·2-s + (−0.341 − 0.939i)3-s − 1.18·4-s − 0.472i·5-s + (1.38 − 0.504i)6-s + 1.84·7-s − 0.272i·8-s + (−0.767 + 0.641i)9-s + 0.697·10-s − 0.599i·11-s + (0.404 + 1.11i)12-s − 1.77·13-s + 2.72i·14-s + (−0.443 + 0.161i)15-s − 0.782·16-s + 0.795i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.341 - 0.939i$
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: \((2,\ 3,\ (\ :34),\ -0.341 - 0.939i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(1.775301480\)
\(L(\frac12)\) \(\approx\) \(1.775301480\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.69e15 + 1.56e16i)T \)
good2 \( 1 - 2.53e10iT - 2.95e20T^{2} \)
5 \( 1 + 2.74e23iT - 3.38e47T^{2} \)
7 \( 1 - 9.96e28T + 2.92e57T^{2} \)
11 \( 1 + 1.53e35iT - 6.52e70T^{2} \)
13 \( 1 + 1.32e38T + 5.59e75T^{2} \)
17 \( 1 - 5.44e41iT - 4.68e83T^{2} \)
19 \( 1 - 2.15e43T + 9.02e86T^{2} \)
23 \( 1 + 1.23e46iT - 3.95e92T^{2} \)
29 \( 1 - 7.71e49iT - 2.77e99T^{2} \)
31 \( 1 + 1.44e50T + 2.58e101T^{2} \)
37 \( 1 - 1.08e52T + 4.34e106T^{2} \)
41 \( 1 + 4.86e53iT - 4.66e109T^{2} \)
43 \( 1 + 1.16e55T + 1.19e111T^{2} \)
47 \( 1 + 1.54e56iT - 5.04e113T^{2} \)
53 \( 1 + 8.65e57iT - 1.78e117T^{2} \)
59 \( 1 + 1.32e60iT - 2.61e120T^{2} \)
61 \( 1 + 4.32e60T + 2.52e121T^{2} \)
67 \( 1 - 1.83e62T + 1.48e124T^{2} \)
71 \( 1 - 1.50e63iT - 7.68e125T^{2} \)
73 \( 1 - 5.26e62T + 5.08e126T^{2} \)
79 \( 1 - 1.62e64T + 1.09e129T^{2} \)
83 \( 1 - 2.51e65iT - 3.14e130T^{2} \)
89 \( 1 - 3.70e66iT - 3.61e132T^{2} \)
97 \( 1 - 1.75e67T + 1.26e135T^{2} \)
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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

−14.12269122462920056477507260883, −12.42649005871637364548116437106, −11.11498929565310190702174343229, −8.623916684948697549618597187355, −7.86636609368719559616398122433, −6.91922038201228514517716907162, −5.34804690391306213994376686533, −4.89439336825401698744895187055, −2.17379826264432260224642104331, −0.942495100710175275915100230521, 0.49873179474224111820284955517, 1.91695276619634861629954453680, 2.90720339100759130036085199672, 4.43791616451691780790325994036, 5.05816170447350255464887327400, 7.47873682654186600074563747815, 9.355727919678966561496563435646, 10.31324856211548560954784636003, 11.42139788758147445829417372905, 12.00295546742677077530790738840

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