Properties

Label 2-3-3.2-c36-0-1
Degree $2$
Conductor $3$
Sign $0.989 - 0.142i$
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  − 4.24e5i·2-s + (3.83e8 − 5.53e7i)3-s − 1.11e11·4-s + 3.68e12i·5-s + (−2.34e13 − 1.62e14i)6-s − 2.28e15·7-s + 1.81e16i·8-s + (1.43e17 − 4.24e16i)9-s + 1.56e18·10-s + 7.84e18i·11-s + (−4.27e19 + 6.16e18i)12-s − 7.34e17·13-s + 9.68e20i·14-s + (2.04e20 + 1.41e21i)15-s + 2.84e19·16-s + 1.60e22i·17-s + ⋯
L(s)  = 1  − 1.61i·2-s + (0.989 − 0.142i)3-s − 1.62·4-s + 0.967i·5-s + (−0.231 − 1.60i)6-s − 1.40·7-s + 1.00i·8-s + (0.959 − 0.282i)9-s + 1.56·10-s + 1.41i·11-s + (−1.60 + 0.231i)12-s − 0.00652·13-s + 2.26i·14-s + (0.138 + 0.957i)15-s + 0.00601·16-s + 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.989 - 0.142i$
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: \((2,\ 3,\ (\ :18),\ 0.989 - 0.142i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(1.510036906\)
\(L(\frac12)\) \(\approx\) \(1.510036906\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.83e8 + 5.53e7i)T \)
good2 \( 1 + 4.24e5iT - 6.87e10T^{2} \)
5 \( 1 - 3.68e12iT - 1.45e25T^{2} \)
7 \( 1 + 2.28e15T + 2.65e30T^{2} \)
11 \( 1 - 7.84e18iT - 3.09e37T^{2} \)
13 \( 1 + 7.34e17T + 1.26e40T^{2} \)
17 \( 1 - 1.60e22iT - 1.97e44T^{2} \)
19 \( 1 - 1.08e23T + 1.08e46T^{2} \)
23 \( 1 - 3.35e24iT - 1.05e49T^{2} \)
29 \( 1 + 1.36e26iT - 4.42e52T^{2} \)
31 \( 1 + 6.89e26T + 4.88e53T^{2} \)
37 \( 1 + 1.67e28T + 2.85e56T^{2} \)
41 \( 1 - 4.32e28iT - 1.14e58T^{2} \)
43 \( 1 + 2.98e29T + 6.38e58T^{2} \)
47 \( 1 + 9.90e29iT - 1.56e60T^{2} \)
53 \( 1 - 1.84e31iT - 1.18e62T^{2} \)
59 \( 1 - 9.96e31iT - 5.63e63T^{2} \)
61 \( 1 - 2.76e31T + 1.87e64T^{2} \)
67 \( 1 - 2.39e31T + 5.47e65T^{2} \)
71 \( 1 + 1.79e33iT - 4.41e66T^{2} \)
73 \( 1 - 1.25e33T + 1.20e67T^{2} \)
79 \( 1 + 4.10e33T + 2.06e68T^{2} \)
83 \( 1 - 4.33e34iT - 1.22e69T^{2} \)
89 \( 1 + 5.33e34iT - 1.50e70T^{2} \)
97 \( 1 + 1.24e35T + 3.34e71T^{2} \)
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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

−18.25299703156596834558251071138, −15.22236030018487298868782374598, −13.47792361089388338532877135970, −12.33604885379951773763619736921, −10.27950761531899793995883630016, −9.457276322049474623693382539633, −7.04977824019934695791902210295, −3.78251511989847344502192631033, −2.91412371371228484820966172743, −1.68259050087227049599615018847, 0.44551898742385936365824221444, 3.30933037888955014482871279920, 5.18752351026199713205434236764, 6.86369801956476344207827240156, 8.468811064670025152917581031039, 9.385270983229128748811199915242, 13.05705990167852680664641484377, 14.13946796418728541204157357229, 16.08718797733363657505176692699, 16.30127509000435380770333615854

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