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Properties

Label 2-6e2-4.3-c32-0-19
Degree $2$
Conductor $36$
Sign $-0.892 + 0.451i$
Analytic cond. $233.519$
Root an. cond. $15.2813$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52e4 + 6.37e4i)2-s + (−3.83e9 + 1.94e9i)4-s + 1.46e11·5-s + 5.71e13i·7-s + (−1.82e14 − 2.14e14i)8-s + (2.23e15 + 9.34e15i)10-s − 6.39e16i·11-s + 3.82e17·13-s + (−3.64e18 + 8.70e17i)14-s + (1.09e19 − 1.48e19i)16-s + 4.07e19·17-s − 6.13e19i·19-s + (−5.61e20 + 2.84e20i)20-s + (4.07e21 − 9.74e20i)22-s + 2.81e20i·23-s + ⋯
L(s)  = 1  + (0.232 + 0.972i)2-s + (−0.892 + 0.451i)4-s + 0.961·5-s + 1.72i·7-s + (−0.646 − 0.762i)8-s + (0.223 + 0.934i)10-s − 1.39i·11-s + 0.574·13-s + (−1.67 + 0.399i)14-s + (0.591 − 0.806i)16-s + 0.837·17-s − 0.212i·19-s + (−0.857 + 0.434i)20-s + (1.35 − 0.323i)22-s + 0.0458i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(233.519\)
Root analytic conductor: \(15.2813\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :16),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.850979214\)
\(L(\frac12)\) \(\approx\) \(1.850979214\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.52e4 - 6.37e4i)T \)
3 \( 1 \)
good5 \( 1 - 1.46e11T + 2.32e22T^{2} \)
7 \( 1 - 5.71e13iT - 1.10e27T^{2} \)
11 \( 1 + 6.39e16iT - 2.11e33T^{2} \)
13 \( 1 - 3.82e17T + 4.42e35T^{2} \)
17 \( 1 - 4.07e19T + 2.36e39T^{2} \)
19 \( 1 + 6.13e19iT - 8.31e40T^{2} \)
23 \( 1 - 2.81e20iT - 3.76e43T^{2} \)
29 \( 1 + 9.24e22T + 6.26e46T^{2} \)
31 \( 1 - 4.23e23iT - 5.29e47T^{2} \)
37 \( 1 + 4.96e23T + 1.52e50T^{2} \)
41 \( 1 + 7.78e25T + 4.06e51T^{2} \)
43 \( 1 - 5.67e24iT - 1.86e52T^{2} \)
47 \( 1 - 2.70e26iT - 3.21e53T^{2} \)
53 \( 1 + 1.23e27T + 1.50e55T^{2} \)
59 \( 1 - 3.43e27iT - 4.64e56T^{2} \)
61 \( 1 - 4.92e28T + 1.35e57T^{2} \)
67 \( 1 - 1.86e29iT - 2.71e58T^{2} \)
71 \( 1 - 7.39e29iT - 1.73e59T^{2} \)
73 \( 1 - 1.01e30T + 4.22e59T^{2} \)
79 \( 1 - 2.63e30iT - 5.29e60T^{2} \)
83 \( 1 + 5.56e30iT - 2.57e61T^{2} \)
89 \( 1 + 1.36e31T + 2.40e62T^{2} \)
97 \( 1 + 9.37e31T + 3.77e63T^{2} \)
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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

−11.49840760374626734991801963278, −9.820317857085693963053686067471, −8.851702516117203698234311991943, −8.238402791691001694930464265020, −6.54552929201726538321324910491, −5.68109588809469247377812643960, −5.34475286153752577268601108909, −3.54187036414617162424224760967, −2.56733451625020619734096250423, −1.16789303810520796092223373374, 0.29512239887453455895957025693, 1.35659402095307148652349477793, 1.98087724309774359962723348264, 3.44092113220169771178831370557, 4.28407313742120187731984516522, 5.34334476313548968078510857745, 6.67000733133857424634176982012, 7.917567128496045568458259792549, 9.592049124829482210810285264816, 10.06494326905437932411455858196

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