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Properties

Label 2-3-3.2-c40-0-6
Degree $2$
Conductor $3$
Sign $0.947 + 0.319i$
Analytic cond. $30.4026$
Root an. cond. $5.51386$
Motivic weight $40$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.62e5i·2-s + (3.30e9 + 1.11e9i)3-s + 5.17e11·4-s + 7.70e13i·5-s + (8.49e14 − 2.52e15i)6-s − 3.01e15·7-s − 1.23e18i·8-s + (9.67e18 + 7.35e18i)9-s + 5.87e19·10-s − 8.18e20i·11-s + (1.71e21 + 5.76e20i)12-s + 1.15e22·13-s + 2.30e21i·14-s + (−8.57e22 + 2.54e23i)15-s − 3.71e23·16-s + 7.23e24i·17-s + ⋯
L(s)  = 1  − 0.727i·2-s + (0.947 + 0.319i)3-s + 0.470·4-s + 0.807i·5-s + (0.232 − 0.689i)6-s − 0.0378·7-s − 1.06i·8-s + (0.796 + 0.605i)9-s + 0.587·10-s − 1.21i·11-s + (0.446 + 0.150i)12-s + 0.606·13-s + 0.0275i·14-s + (−0.257 + 0.765i)15-s − 0.307·16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(30.4026\)
Root analytic conductor: \(5.51386\)
Motivic weight: \(40\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :20),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(3.609315589\)
\(L(\frac12)\) \(\approx\) \(3.609315589\)
\(L(21)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.30e9 - 1.11e9i)T \)
good2 \( 1 + 7.62e5iT - 1.09e12T^{2} \)
5 \( 1 - 7.70e13iT - 9.09e27T^{2} \)
7 \( 1 + 3.01e15T + 6.36e33T^{2} \)
11 \( 1 + 8.18e20iT - 4.52e41T^{2} \)
13 \( 1 - 1.15e22T + 3.61e44T^{2} \)
17 \( 1 - 7.23e24iT - 1.65e49T^{2} \)
19 \( 1 - 2.86e25T + 1.41e51T^{2} \)
23 \( 1 + 7.48e25iT - 2.94e54T^{2} \)
29 \( 1 - 1.79e28iT - 3.13e58T^{2} \)
31 \( 1 - 8.58e29T + 4.51e59T^{2} \)
37 \( 1 - 2.97e31T + 5.34e62T^{2} \)
41 \( 1 + 3.27e31iT - 3.24e64T^{2} \)
43 \( 1 + 7.95e32T + 2.18e65T^{2} \)
47 \( 1 - 2.08e33iT - 7.65e66T^{2} \)
53 \( 1 + 5.75e34iT - 9.35e68T^{2} \)
59 \( 1 - 7.76e34iT - 6.82e70T^{2} \)
61 \( 1 - 3.24e35T + 2.58e71T^{2} \)
67 \( 1 + 2.79e36T + 1.10e73T^{2} \)
71 \( 1 - 1.05e37iT - 1.12e74T^{2} \)
73 \( 1 - 6.07e36T + 3.41e74T^{2} \)
79 \( 1 - 3.31e37T + 8.03e75T^{2} \)
83 \( 1 + 9.06e37iT - 5.79e76T^{2} \)
89 \( 1 + 1.59e39iT - 9.45e77T^{2} \)
97 \( 1 + 3.49e39T + 2.95e79T^{2} \)
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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

−16.14129130043188593130000845101, −14.78489367739392797868153881737, −13.22469716144708425548989455165, −11.19836690264097479760006523693, −10.12262757246797233225688963342, −8.242158755678304786001227022258, −6.43430035369791475847584786789, −3.67600596914215434811654955016, −2.84707087765977643868322808856, −1.36170011714590443105302010600, 1.23675663792823356286318098843, 2.71050759055928050243392939116, 4.83406198984756890879140075740, 6.83049558042332470938986684236, 8.001262811555019229010493321787, 9.479022796898590458109695433560, 11.98874764385454927965140643764, 13.63870773416061714954106827838, 15.15154695211595856614977514413, 16.30148401129433822652246727147

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