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Properties

Label 2-6e2-12.11-c1-0-1
Degree $2$
Conductor $36$
Sign $0.577 + 0.816i$
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  − 1.41i·2-s − 2.00·4-s + 1.41i·5-s + 2.82i·8-s + 2.00·10-s − 4·13-s + 4.00·16-s − 7.07i·17-s − 2.82i·20-s + 2.99·25-s + 5.65i·26-s + 9.89i·29-s − 5.65i·32-s − 10.0·34-s + 2·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 0.632i·5-s + 1.00i·8-s + 0.632·10-s − 1.10·13-s + 1.00·16-s − 1.71i·17-s − 0.632i·20-s + 0.599·25-s + 1.10i·26-s + 1.83i·29-s − 1.00i·32-s − 1.71·34-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611374 - 0.316470i\)
\(L(\frac12)\) \(\approx\) \(0.611374 - 0.316470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 7.07iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.3iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−16.52648936978600694773576605124, −14.80336869328748637992644873152, −13.92483647868638341386564708123, −12.54150389996348553379979819090, −11.44800817633426214628243650161, −10.27676678139585969768072929358, −9.103835090281441945583940890664, −7.26184237017286218553976308465, −4.97377548933302703236598212025, −2.88466240953240789052920623905, 4.43545401389281113981475638487, 5.98621674530661777629858810811, 7.64417032236982256320859363833, 8.855499245950922138095638970351, 10.18182168954013904720347049411, 12.26445635236430275399076339267, 13.25198629763988668194638008179, 14.60737384875976943843829185846, 15.50835549989476606137386753619, 16.87136183962286060548292253966

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