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Properties

Label 2-6e2-36.11-c1-0-0
Degree $2$
Conductor $36$
Sign $0.892 - 0.451i$
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  + (−1.41 − 0.0786i)2-s + (0.637 + 1.61i)3-s + (1.98 + 0.222i)4-s + (0.686 − 0.396i)5-s + (−0.774 − 2.32i)6-s + (−2.35 − 1.35i)7-s + (−2.78 − 0.469i)8-s + (−2.18 + 2.05i)9-s + (−0.999 + 0.505i)10-s + (1.71 − 2.96i)11-s + (0.910 + 3.34i)12-s + (−1.68 − 2.92i)13-s + (3.21 + 2.10i)14-s + (1.07 + 0.852i)15-s + (3.90 + 0.882i)16-s + 2.52i·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0556i)2-s + (0.368 + 0.929i)3-s + (0.993 + 0.111i)4-s + (0.306 − 0.177i)5-s + (−0.316 − 0.948i)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 + (0.262 + 0.964i)12-s + (−0.467 − 0.809i)13-s + (0.858 + 0.561i)14-s + (0.277 + 0.220i)15-s + (0.975 + 0.220i)16-s + 0.612i·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539566 + 0.128772i\)
\(L(\frac12)\) \(\approx\) \(0.539566 + 0.128772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0786i)T \)
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

−16.66969579784214730498949882954, −15.78783725616046755500188532613, −14.57156996359569565933870937295, −13.02102111206209898506286307331, −11.24510184214291654017450503633, −10.10775794044941145069587435610, −9.289588685598248005923836495337, −7.927550193705076381336876952662, −6.00136149408027008257477790444, −3.35888230345847000471252667227, 2.44171528296503042666414328994, 6.34259575125884772498092378583, 7.25147504858081772664063154981, 8.930884703281486522628767796269, 9.784983529081480783211548529942, 11.68604662922207711970596641855, 12.60065656763258940379422092608, 14.18010902533107633907303837178, 15.36727123283390770636544800957, 16.73210382029695968105843262199

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