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Properties

Label 2-6e2-4.3-c32-0-62
Degree $2$
Conductor $36$
Sign $-0.848 - 0.529i$
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  + (1.80e4 − 6.29e4i)2-s + (−3.64e9 − 2.27e9i)4-s − 1.21e11·5-s − 3.17e10i·7-s + (−2.09e14 + 1.88e14i)8-s + (−2.18e15 + 7.63e15i)10-s − 6.68e16i·11-s + 5.92e17·13-s + (−1.99e15 − 5.72e14i)14-s + (8.09e18 + 1.65e19i)16-s − 4.82e19·17-s − 9.57e19i·19-s + (4.41e20 + 2.75e20i)20-s + (−4.21e21 − 1.20e21i)22-s + 5.33e20i·23-s + ⋯
L(s)  = 1  + (0.275 − 0.961i)2-s + (−0.848 − 0.529i)4-s − 0.794·5-s − 0.000954i·7-s + (−0.742 + 0.669i)8-s + (−0.218 + 0.763i)10-s − 1.45i·11-s + 0.889·13-s + (−0.000917 − 0.000263i)14-s + (0.438 + 0.898i)16-s − 0.992·17-s − 0.331i·19-s + (0.673 + 0.421i)20-s + (−1.39 − 0.400i)22-s + 0.0869i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.848 - 0.529i$
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: \((2,\ 36,\ (\ :16),\ -0.848 - 0.529i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.377893577\)
\(L(\frac12)\) \(\approx\) \(1.377893577\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.80e4 + 6.29e4i)T \)
3 \( 1 \)
good5 \( 1 + 1.21e11T + 2.32e22T^{2} \)
7 \( 1 + 3.17e10iT - 1.10e27T^{2} \)
11 \( 1 + 6.68e16iT - 2.11e33T^{2} \)
13 \( 1 - 5.92e17T + 4.42e35T^{2} \)
17 \( 1 + 4.82e19T + 2.36e39T^{2} \)
19 \( 1 + 9.57e19iT - 8.31e40T^{2} \)
23 \( 1 - 5.33e20iT - 3.76e43T^{2} \)
29 \( 1 - 4.06e23T + 6.26e46T^{2} \)
31 \( 1 - 4.12e23iT - 5.29e47T^{2} \)
37 \( 1 - 1.13e25T + 1.52e50T^{2} \)
41 \( 1 - 2.76e23T + 4.06e51T^{2} \)
43 \( 1 + 1.68e26iT - 1.86e52T^{2} \)
47 \( 1 + 7.88e26iT - 3.21e53T^{2} \)
53 \( 1 - 5.53e27T + 1.50e55T^{2} \)
59 \( 1 + 4.25e28iT - 4.64e56T^{2} \)
61 \( 1 - 5.00e28T + 1.35e57T^{2} \)
67 \( 1 - 1.33e29iT - 2.71e58T^{2} \)
71 \( 1 + 5.68e29iT - 1.73e59T^{2} \)
73 \( 1 + 6.17e29T + 4.22e59T^{2} \)
79 \( 1 + 1.44e30iT - 5.29e60T^{2} \)
83 \( 1 - 6.98e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.07e31T + 2.40e62T^{2} \)
97 \( 1 + 3.35e31T + 3.77e63T^{2} \)
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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

−10.41528404283800496299766540933, −8.860086952949654998352301293828, −8.299312084709050695779487939690, −6.54643527428625248458117495723, −5.38985543489131254496605828547, −4.12682590156424892370721844162, −3.43002468989799757969790703185, −2.34723229376907078804220974631, −0.935172790955911033595409964134, −0.30931442121464625536319499503, 0.990275197684671596046007328124, 2.61416575170829676364370102877, 4.07986103876724545174510778359, 4.50152618504532957527658103619, 5.96224793415634875544183260786, 6.97442321756749859296906431645, 7.86739408375511821895018189819, 8.826199468604781853886782548806, 10.03917624166267096286648253619, 11.56281458739190251580922179164

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