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Properties

Label 2-3-3.2-c34-0-2
Degree $2$
Conductor $3$
Sign $0.631 - 0.775i$
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  − 5.93e4i·2-s + (−8.15e7 + 1.00e8i)3-s + 1.36e10·4-s + 7.02e10i·5-s + (5.94e12 + 4.83e12i)6-s + 9.28e13·7-s − 1.82e15i·8-s + (−3.39e15 − 1.63e16i)9-s + 4.16e15·10-s + 4.29e17i·11-s + (−1.11e18 + 1.36e18i)12-s − 3.39e18·13-s − 5.50e18i·14-s + (−7.03e18 − 5.72e18i)15-s + 1.26e20·16-s + 2.02e20i·17-s + ⋯
L(s)  = 1  − 0.452i·2-s + (−0.631 + 0.775i)3-s + 0.795·4-s + 0.0920i·5-s + (0.351 + 0.285i)6-s + 0.399·7-s − 0.812i·8-s + (−0.203 − 0.979i)9-s + 0.0416·10-s + 0.850i·11-s + (−0.501 + 0.616i)12-s − 0.392·13-s − 0.180i·14-s + (−0.0714 − 0.0580i)15-s + 0.427·16-s + 0.244i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.631 - 0.775i$
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: \((2,\ 3,\ (\ :17),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.70326 + 0.809996i\)
\(L(\frac12)\) \(\approx\) \(1.70326 + 0.809996i\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.15e7 - 1.00e8i)T \)
good2 \( 1 + 5.93e4iT - 1.71e10T^{2} \)
5 \( 1 - 7.02e10iT - 5.82e23T^{2} \)
7 \( 1 - 9.28e13T + 5.41e28T^{2} \)
11 \( 1 - 4.29e17iT - 2.55e35T^{2} \)
13 \( 1 + 3.39e18T + 7.48e37T^{2} \)
17 \( 1 - 2.02e20iT - 6.84e41T^{2} \)
19 \( 1 - 6.64e21T + 3.00e43T^{2} \)
23 \( 1 - 2.42e23iT - 1.98e46T^{2} \)
29 \( 1 - 1.27e25iT - 5.26e49T^{2} \)
31 \( 1 + 2.14e23T + 5.08e50T^{2} \)
37 \( 1 + 5.46e26T + 2.08e53T^{2} \)
41 \( 1 - 3.09e27iT - 6.83e54T^{2} \)
43 \( 1 - 4.94e27T + 3.45e55T^{2} \)
47 \( 1 + 7.73e26iT - 7.10e56T^{2} \)
53 \( 1 + 2.24e29iT - 4.22e58T^{2} \)
59 \( 1 - 1.72e30iT - 1.61e60T^{2} \)
61 \( 1 - 2.54e30T + 5.02e60T^{2} \)
67 \( 1 - 1.55e30T + 1.22e62T^{2} \)
71 \( 1 + 3.31e31iT - 8.76e62T^{2} \)
73 \( 1 - 6.69e31T + 2.25e63T^{2} \)
79 \( 1 - 1.59e32T + 3.30e64T^{2} \)
83 \( 1 + 2.64e31iT - 1.77e65T^{2} \)
89 \( 1 + 1.26e33iT - 1.90e66T^{2} \)
97 \( 1 + 2.38e33T + 3.55e67T^{2} \)
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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

−17.79025969862534538774460395317, −16.20468139730118743036298114355, −14.91239018844976829006064201233, −12.24748381801643019967539482522, −11.05062539421376276944777181968, −9.716724165375390444839170033947, −7.07272308097595168216589293565, −5.18060825251202171323770788410, −3.32358755365220778547994514594, −1.38298227538946196601069688092, 0.77132293603283925286359200302, 2.44020525752117183116849283726, 5.32927182373709163889044225690, 6.70638167995126264220315821674, 8.072876447092187621371355987031, 10.91091132589439230230999255456, 12.14798149438080716653460290337, 14.11692084298171102960591539281, 16.07547857515082401145175132316, 17.26762828415570411792401737675

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