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Properties

Label 2-6e2-36.23-c1-0-3
Degree $2$
Conductor $36$
Sign $0.773 + 0.633i$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 − 1.37i)2-s + (1.35 + 1.07i)3-s + (−1.77 + 0.922i)4-s + (−2.18 − 1.26i)5-s + (1.02 − 2.22i)6-s + (−1.10 + 0.637i)7-s + (1.86 + 2.12i)8-s + (0.686 + 2.92i)9-s + (−1 + 3.42i)10-s + (−0.252 − 0.437i)11-s + (−3.40 − 0.656i)12-s + (1.18 − 2.05i)13-s + (1.24 + 1.30i)14-s + (−1.61 − 4.06i)15-s + (2.29 − 3.27i)16-s + 0.792i·17-s + ⋯
L(s)  = 1  + (−0.237 − 0.971i)2-s + (0.783 + 0.621i)3-s + (−0.887 + 0.461i)4-s + (−0.977 − 0.564i)5-s + (0.417 − 0.908i)6-s + (−0.417 + 0.241i)7-s + (0.658 + 0.752i)8-s + (0.228 + 0.973i)9-s + (−0.316 + 1.08i)10-s + (−0.0761 − 0.131i)11-s + (−0.981 − 0.189i)12-s + (0.328 − 0.569i)13-s + (0.333 + 0.348i)14-s + (−0.415 − 1.04i)15-s + (0.574 − 0.818i)16-s + 0.192i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647616 - 0.231324i\)
\(L(\frac12)\) \(\approx\) \(0.647616 - 0.231324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.335 + 1.37i)T \)
3 \( 1 + (-1.35 - 1.07i)T \)
good5 \( 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.10 - 0.637i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.252 + 0.437i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.18 + 2.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.792iT - 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 + (1.61 - 2.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.18 + 1.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.04 - 4.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + (-5.87 - 3.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.69 - 3.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.599 + 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.87iT - 53T^{2} \)
59 \( 1 + (-6.18 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 2.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 + 3.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 + (-8.55 + 4.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.82 - 6.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + (5.24 + 9.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.20527252763093359641639356516, −15.38556049322962879914970623083, −13.79181005060571725737888296180, −12.71910696071558134793136109201, −11.45110148393896745343517263190, −10.13762615039824832070604494286, −8.869941070044186185832435225293, −7.975639787693418735192102406061, −4.66978679232252476157993146347, −3.21460605818579881970173812500, 3.85338976274152635267506907552, 6.49847748207673495252183723233, 7.53554169894284660944570720392, 8.599505684992496868552290190282, 10.11435536820438041781587660080, 12.05528771447840144100721012862, 13.48406215298052572986837508358, 14.46358948403521602351530929536, 15.40919858884089810576011753783, 16.42968354523109396719367623473

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