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Properties

Label 2-3-3.2-c34-0-1
Degree $2$
Conductor $3$
Sign $0.891 + 0.453i$
Analytic cond. $21.9676$
Root an. cond. $4.68697$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.03e5i·2-s + (−1.15e8 − 5.85e7i)3-s − 2.43e10·4-s − 1.79e11i·5-s + (−1.19e13 + 2.34e13i)6-s − 2.97e14·7-s + 1.46e15i·8-s + (9.81e15 + 1.34e16i)9-s − 3.66e16·10-s + 5.45e17i·11-s + (2.80e18 + 1.42e18i)12-s + 6.76e18·13-s + 6.07e19i·14-s + (−1.05e19 + 2.07e19i)15-s − 1.19e20·16-s − 1.38e21i·17-s + ⋯
L(s)  = 1  − 1.55i·2-s + (−0.891 − 0.453i)3-s − 1.41·4-s − 0.235i·5-s + (−0.705 + 1.38i)6-s − 1.28·7-s + 0.651i·8-s + (0.588 + 0.808i)9-s − 0.366·10-s + 1.07i·11-s + (1.26 + 0.643i)12-s + 0.781·13-s + 1.99i·14-s + (−0.106 + 0.210i)15-s − 0.405·16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(21.9676\)
Root analytic conductor: \(4.68697\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :17),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.419620 - 0.100623i\)
\(L(\frac12)\) \(\approx\) \(0.419620 - 0.100623i\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15e8 + 5.85e7i)T \)
good2 \( 1 + 2.03e5iT - 1.71e10T^{2} \)
5 \( 1 + 1.79e11iT - 5.82e23T^{2} \)
7 \( 1 + 2.97e14T + 5.41e28T^{2} \)
11 \( 1 - 5.45e17iT - 2.55e35T^{2} \)
13 \( 1 - 6.76e18T + 7.48e37T^{2} \)
17 \( 1 + 1.38e21iT - 6.84e41T^{2} \)
19 \( 1 + 8.87e21T + 3.00e43T^{2} \)
23 \( 1 + 2.99e22iT - 1.98e46T^{2} \)
29 \( 1 - 3.17e24iT - 5.26e49T^{2} \)
31 \( 1 - 1.17e24T + 5.08e50T^{2} \)
37 \( 1 - 3.74e26T + 2.08e53T^{2} \)
41 \( 1 - 2.13e27iT - 6.83e54T^{2} \)
43 \( 1 - 4.08e27T + 3.45e55T^{2} \)
47 \( 1 - 4.95e28iT - 7.10e56T^{2} \)
53 \( 1 + 9.35e28iT - 4.22e58T^{2} \)
59 \( 1 - 6.32e29iT - 1.61e60T^{2} \)
61 \( 1 - 4.42e29T + 5.02e60T^{2} \)
67 \( 1 + 4.63e30T + 1.22e62T^{2} \)
71 \( 1 - 2.74e30iT - 8.76e62T^{2} \)
73 \( 1 + 2.72e31T + 2.25e63T^{2} \)
79 \( 1 - 1.93e32T + 3.30e64T^{2} \)
83 \( 1 - 2.76e32iT - 1.77e65T^{2} \)
89 \( 1 - 2.71e33iT - 1.90e66T^{2} \)
97 \( 1 + 1.23e33T + 3.55e67T^{2} \)
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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

−18.12165940898761374134857587168, −16.24686613418711886987030719898, −13.13244291643349824051290245561, −12.37281268256853459470967607576, −10.87138921462987119474945380601, −9.511824322907379046483268317199, −6.65702022573582472208214846402, −4.48371541581384473522135419469, −2.59430374834434561190921844322, −0.999239915956407417921351231827, 0.21421705085980024513510636907, 3.88410365563059883323344267069, 5.92408995724754477732184215204, 6.50285334185561625836501982813, 8.705713772228650130568670574395, 10.66774744121030656997684433348, 13.05911987448929813966780220463, 15.08143227566207740305705129236, 16.24079333731649413150294533679, 17.11680614282279556899443799517

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