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Properties

Label 2-6e2-12.11-c3-0-0
Degree $2$
Conductor $36$
Sign $0.109 - 0.993i$
Analytic cond. $2.12406$
Root an. cond. $1.45741$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.73 + 0.707i)2-s + (7 − 3.87i)4-s + 9.89i·5-s + 30.9i·7-s + (−16.4 + 15.5i)8-s + (−7.00 − 27.1i)10-s + 43.8·11-s − 28·13-s + (−21.9 − 84.8i)14-s + (34 − 54.2i)16-s − 49.4i·17-s + (38.3 + 69.2i)20-s + (−120 + 30.9i)22-s − 43.8·23-s + 27·25-s + (76.6 − 19.7i)26-s + ⋯
L(s)  = 1  + (−0.968 + 0.249i)2-s + (0.875 − 0.484i)4-s + 0.885i·5-s + 1.67i·7-s + (−0.726 + 0.687i)8-s + (−0.221 − 0.857i)10-s + 1.20·11-s − 0.597·13-s + (−0.418 − 1.61i)14-s + (0.531 − 0.847i)16-s − 0.706i·17-s + (0.428 + 0.774i)20-s + (−1.16 + 0.300i)22-s − 0.397·23-s + 0.215·25-s + (0.578 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ 0.109 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.610347 + 0.546583i\)
\(L(\frac12)\) \(\approx\) \(0.610347 + 0.546583i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.73 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 - 9.89iT - 125T^{2} \)
7 \( 1 - 30.9iT - 343T^{2} \)
11 \( 1 - 43.8T + 1.33e3T^{2} \)
13 \( 1 + 28T + 2.19e3T^{2} \)
17 \( 1 + 49.4iT - 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 43.8T + 1.21e4T^{2} \)
29 \( 1 - 137. iT - 2.43e4T^{2} \)
31 \( 1 + 216. iT - 2.97e4T^{2} \)
37 \( 1 - 266T + 5.06e4T^{2} \)
41 \( 1 + 227. iT - 6.89e4T^{2} \)
43 \( 1 + 61.9iT - 7.95e4T^{2} \)
47 \( 1 - 306.T + 1.03e5T^{2} \)
53 \( 1 - 120. iT - 1.48e5T^{2} \)
59 \( 1 - 613.T + 2.05e5T^{2} \)
61 \( 1 - 350T + 2.26e5T^{2} \)
67 \( 1 - 433. iT - 3.00e5T^{2} \)
71 \( 1 + 657.T + 3.57e5T^{2} \)
73 \( 1 + 112T + 3.89e5T^{2} \)
79 \( 1 - 216. iT - 4.93e5T^{2} \)
83 \( 1 - 306.T + 5.71e5T^{2} \)
89 \( 1 + 1.29e3iT - 7.04e5T^{2} \)
97 \( 1 + 616T + 9.12e5T^{2} \)
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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

−16.28131807117205648574945736803, −15.09091694038472564298254589483, −14.47894587106254844761977138517, −12.14916284928116359643821543396, −11.31146878525343081813341238039, −9.722924883994673495355079105488, −8.761826211172650942215836850472, −7.09828321479938178490338792318, −5.82885073410147353573749123308, −2.51366582009756001478481545201, 1.06914759206160677010192137974, 4.08882828130530318493618473588, 6.72521120562218333706781789771, 8.044245647023103132508827631965, 9.417704547852365393817889860609, 10.52359794268058887017911781833, 11.86641333264027639645832440866, 13.09139185187255148533392603923, 14.56891429824734552406390273594, 16.31234659830975255981627904751

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