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Properties

Label 2-3-3.2-c36-0-4
Degree $2$
Conductor $3$
Sign $0.199 - 0.979i$
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  + 1.84e5i·2-s + (7.73e7 − 3.79e8i)3-s + 3.47e10·4-s + 1.78e12i·5-s + (6.99e13 + 1.42e13i)6-s − 1.12e15·7-s + 1.90e16i·8-s + (−1.38e17 − 5.87e16i)9-s − 3.29e17·10-s + 3.23e18i·11-s + (2.68e18 − 1.31e19i)12-s + 1.98e20·13-s − 2.06e20i·14-s + (6.79e20 + 1.38e20i)15-s − 1.12e21·16-s + 9.74e21i·17-s + ⋯
L(s)  = 1  + 0.702i·2-s + (0.199 − 0.979i)3-s + 0.505·4-s + 0.468i·5-s + (0.688 + 0.140i)6-s − 0.688·7-s + 1.05i·8-s + (−0.920 − 0.391i)9-s − 0.329·10-s + 0.581i·11-s + (0.100 − 0.495i)12-s + 1.76·13-s − 0.483i·14-s + (0.459 + 0.0936i)15-s − 0.238·16-s + 0.693i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.199 - 0.979i$
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: \((2,\ 3,\ (\ :18),\ 0.199 - 0.979i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(2.195141047\)
\(L(\frac12)\) \(\approx\) \(2.195141047\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.73e7 + 3.79e8i)T \)
good2 \( 1 - 1.84e5iT - 6.87e10T^{2} \)
5 \( 1 - 1.78e12iT - 1.45e25T^{2} \)
7 \( 1 + 1.12e15T + 2.65e30T^{2} \)
11 \( 1 - 3.23e18iT - 3.09e37T^{2} \)
13 \( 1 - 1.98e20T + 1.26e40T^{2} \)
17 \( 1 - 9.74e21iT - 1.97e44T^{2} \)
19 \( 1 + 4.96e22T + 1.08e46T^{2} \)
23 \( 1 - 1.29e24iT - 1.05e49T^{2} \)
29 \( 1 - 3.60e26iT - 4.42e52T^{2} \)
31 \( 1 - 7.63e26T + 4.88e53T^{2} \)
37 \( 1 + 3.30e27T + 2.85e56T^{2} \)
41 \( 1 + 1.75e28iT - 1.14e58T^{2} \)
43 \( 1 - 2.19e29T + 6.38e58T^{2} \)
47 \( 1 - 2.28e30iT - 1.56e60T^{2} \)
53 \( 1 + 8.49e30iT - 1.18e62T^{2} \)
59 \( 1 + 1.12e32iT - 5.63e63T^{2} \)
61 \( 1 - 1.74e32T + 1.87e64T^{2} \)
67 \( 1 + 2.93e32T + 5.47e65T^{2} \)
71 \( 1 + 3.75e32iT - 4.41e66T^{2} \)
73 \( 1 + 2.87e33T + 1.20e67T^{2} \)
79 \( 1 + 2.55e34T + 2.06e68T^{2} \)
83 \( 1 - 2.12e34iT - 1.22e69T^{2} \)
89 \( 1 + 1.66e35iT - 1.50e70T^{2} \)
97 \( 1 - 3.09e35T + 3.34e71T^{2} \)
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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

−17.62315002165663037594105411121, −15.93287055408828465733877422647, −14.43659876366898448426984245389, −12.79949046164589121048909783179, −11.00366876791826202432344049146, −8.461001195518561561699029244224, −6.92748708717130075901044081744, −6.08037648312173349275967751240, −3.09616954905750847359994087091, −1.50188740415607498250490674151, 0.74560183464791159840684000864, 2.76870862969677278276340936532, 3.99235354564212117553334177222, 6.12796317765100332643573179898, 8.716574034597414170109186056159, 10.24723010364294076640100718464, 11.51159187926586159257501844078, 13.39378796997355998004186491211, 15.62864091917331644375042828976, 16.47960545482259638677683463237

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