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Properties

Label 2-6e2-36.23-c1-0-0
Degree $2$
Conductor $36$
Sign $0.166 - 0.986i$
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  + (−0.774 + 1.18i)2-s + (−0.637 + 1.61i)3-s + (−0.801 − 1.83i)4-s + (0.686 + 0.396i)5-s + (−1.41 − 2.00i)6-s + (2.35 − 1.35i)7-s + (2.78 + 0.469i)8-s + (−2.18 − 2.05i)9-s + (−1 + 0.505i)10-s + (−1.71 − 2.96i)11-s + (3.46 − 0.121i)12-s + (−1.68 + 2.92i)13-s + (−0.213 + 3.83i)14-s + (−1.07 + 0.852i)15-s + (−2.71 + 2.93i)16-s − 2.52i·17-s + ⋯
L(s)  = 1  + (−0.547 + 0.836i)2-s + (−0.368 + 0.929i)3-s + (−0.400 − 0.916i)4-s + (0.306 + 0.177i)5-s + (−0.576 − 0.817i)6-s + (0.888 − 0.513i)7-s + (0.986 + 0.166i)8-s + (−0.728 − 0.684i)9-s + (−0.316 + 0.159i)10-s + (−0.516 − 0.894i)11-s + (0.999 − 0.0351i)12-s + (−0.467 + 0.809i)13-s + (−0.0570 + 1.02i)14-s + (−0.277 + 0.220i)15-s + (−0.678 + 0.734i)16-s − 0.612i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.166 - 0.986i$
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: \((2,\ 36,\ (\ :1/2),\ 0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427753 + 0.361501i\)
\(L(\frac12)\) \(\approx\) \(0.427753 + 0.361501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.774 - 1.18i)T \)
3 \( 1 + (0.637 - 1.61i)T \)
good5 \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.35 + 1.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 - 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 - 2.20iT - 19T^{2} \)
23 \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 - 0.396i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.47 + 0.852i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + (-0.127 - 0.0737i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.01 - 3.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.77 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.51iT - 53T^{2} \)
59 \( 1 + (-2.58 + 4.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.01 + 3.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (-8.80 + 5.08i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.34iT - 89T^{2} \)
97 \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.70985012382218553682260935486, −15.88673281793730288996005956496, −14.56503076456082401959329297435, −13.87243004290686964766825119737, −11.44385518035936395695076494494, −10.42897571371634572352597442852, −9.267143690196014933793692168026, −7.81937093685503095898891737418, −6.02210048266719320367213666561, −4.62632618248830518501790350783, 2.12226860007967066366062149714, 5.18938050780882079827077089902, 7.43637056954322732575378759682, 8.534285118964387069443086391372, 10.22233359038497806288468788253, 11.49255145271192746749948920204, 12.50552586011408585776297769771, 13.36656999084686644948595551036, 14.99755678490269295429074183774, 16.89506192121168170823699174585

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