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Properties

Label 2-3-3.2-c36-0-3
Degree $2$
Conductor $3$
Sign $-0.775 - 0.631i$
Analytic cond. $24.6273$
Root an. cond. $4.96259$
Motivic weight $36$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.81e5i·2-s + (−3.00e8 − 2.44e8i)3-s − 1.62e11·4-s + 1.10e12i·5-s + (1.17e14 − 1.44e14i)6-s + 2.90e15·7-s − 4.53e16i·8-s + (3.04e16 + 1.46e17i)9-s − 5.33e17·10-s − 4.36e18i·11-s + (4.89e19 + 3.98e19i)12-s + 4.74e19·13-s + 1.39e21i·14-s + (2.71e20 − 3.33e20i)15-s + 1.06e22·16-s − 5.47e21i·17-s + ⋯
L(s)  = 1  + 1.83i·2-s + (−0.775 − 0.631i)3-s − 2.37·4-s + 0.290i·5-s + (1.15 − 1.42i)6-s + 1.78·7-s − 2.51i·8-s + (0.203 + 0.979i)9-s − 0.533·10-s − 0.785i·11-s + (1.83 + 1.49i)12-s + 0.422·13-s + 3.27i·14-s + (0.183 − 0.225i)15-s + 2.24·16-s − 0.389i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.775 - 0.631i$
Analytic conductor: \(24.6273\)
Root analytic conductor: \(4.96259\)
Motivic weight: \(36\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :18),\ -0.775 - 0.631i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(1.460203045\)
\(L(\frac12)\) \(\approx\) \(1.460203045\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.00e8 + 2.44e8i)T \)
good2 \( 1 - 4.81e5iT - 6.87e10T^{2} \)
5 \( 1 - 1.10e12iT - 1.45e25T^{2} \)
7 \( 1 - 2.90e15T + 2.65e30T^{2} \)
11 \( 1 + 4.36e18iT - 3.09e37T^{2} \)
13 \( 1 - 4.74e19T + 1.26e40T^{2} \)
17 \( 1 + 5.47e21iT - 1.97e44T^{2} \)
19 \( 1 + 3.53e22T + 1.08e46T^{2} \)
23 \( 1 - 3.93e24iT - 1.05e49T^{2} \)
29 \( 1 - 8.94e25iT - 4.42e52T^{2} \)
31 \( 1 + 2.09e26T + 4.88e53T^{2} \)
37 \( 1 - 1.24e28T + 2.85e56T^{2} \)
41 \( 1 - 1.88e29iT - 1.14e58T^{2} \)
43 \( 1 + 3.88e28T + 6.38e58T^{2} \)
47 \( 1 + 3.19e28iT - 1.56e60T^{2} \)
53 \( 1 - 7.64e30iT - 1.18e62T^{2} \)
59 \( 1 - 7.04e31iT - 5.63e63T^{2} \)
61 \( 1 - 2.17e32T + 1.87e64T^{2} \)
67 \( 1 + 9.44e32T + 5.47e65T^{2} \)
71 \( 1 - 7.27e32iT - 4.41e66T^{2} \)
73 \( 1 - 4.28e32T + 1.20e67T^{2} \)
79 \( 1 - 6.42e32T + 2.06e68T^{2} \)
83 \( 1 + 4.10e34iT - 1.22e69T^{2} \)
89 \( 1 - 1.85e35iT - 1.50e70T^{2} \)
97 \( 1 - 3.60e35T + 3.34e71T^{2} \)
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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

−17.67574828694939073601482170744, −16.44766340403157005721069439073, −14.75669609015447166208474602289, −13.54523533477767418024856089252, −11.23275579445539802287236547057, −8.408298821867460536964198511131, −7.33628738073702511507377527924, −5.85584813580425268183301911867, −4.75242817224134003750730303875, −1.08991605284381146916771936616, 0.72638315059615559206983364755, 1.96188350160833035899264126579, 4.14471201839254364311439302915, 5.00443866528579389921562980745, 8.697088927832679286020836414036, 10.40353335313745704852179637828, 11.34554665276769517258538211928, 12.52520376518466753750192393853, 14.62091172529024793314939476270, 17.32105784946924560769465681900

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