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Properties

Label 2-6e2-9.5-c4-0-3
Degree $2$
Conductor $36$
Sign $-0.395 + 0.918i$
Analytic cond. $3.72131$
Root an. cond. $1.92907$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.256 − 8.99i)3-s + (−7.67 − 4.43i)5-s + (−30.9 − 53.5i)7-s + (−80.8 − 4.61i)9-s + (94.7 − 54.7i)11-s + (77.8 − 134. i)13-s + (−41.8 + 67.9i)15-s + 395. i·17-s + 140.·19-s + (−490. + 264. i)21-s + (802. + 463. i)23-s + (−273. − 473. i)25-s + (−62.3 + 726. i)27-s + (323. − 186. i)29-s + (521. − 903. i)31-s + ⋯
L(s)  = 1  + (0.0285 − 0.999i)3-s + (−0.307 − 0.177i)5-s + (−0.631 − 1.09i)7-s + (−0.998 − 0.0570i)9-s + (0.783 − 0.452i)11-s + (0.460 − 0.798i)13-s + (−0.186 + 0.302i)15-s + 1.37i·17-s + 0.388·19-s + (−1.11 + 0.599i)21-s + (1.51 + 0.876i)23-s + (−0.437 − 0.757i)25-s + (−0.0854 + 0.996i)27-s + (0.385 − 0.222i)29-s + (0.543 − 0.940i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(3.72131\)
Root analytic conductor: \(1.92907\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :2),\ -0.395 + 0.918i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.658627 - 1.00016i\)
\(L(\frac12)\) \(\approx\) \(0.658627 - 1.00016i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.256 + 8.99i)T \)
good5 \( 1 + (7.67 + 4.43i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (30.9 + 53.5i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-94.7 + 54.7i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-77.8 + 134. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 395. iT - 8.35e4T^{2} \)
19 \( 1 - 140.T + 1.30e5T^{2} \)
23 \( 1 + (-802. - 463. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-323. + 186. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-521. + 903. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 194.T + 1.87e6T^{2} \)
41 \( 1 + (2.34e3 + 1.35e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (167. + 290. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.46e3 + 1.42e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.76e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.35e3 + 2.51e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-3.52e3 - 6.10e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.43e3 - 5.95e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 821. iT - 2.54e7T^{2} \)
73 \( 1 - 4.09e3T + 2.83e7T^{2} \)
79 \( 1 + (3.78e3 + 6.55e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-6.77e3 + 3.91e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.28e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.89e3 - 3.27e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.28433512690721603302267058275, −13.78921953381448328523581287282, −13.08456786798556711971891271953, −11.81412770996647847153293016213, −10.46949619308018215256541617667, −8.650287872900836555957864606195, −7.36296222142253873411879556290, −6.08755526591299637991715175892, −3.58584229886317101912470095279, −0.897499615307426482898906792143, 3.12279790009191266401488123388, 4.92039720834344289734560014032, 6.65624242790857659934643786170, 8.851058862585644979385790173690, 9.582478011906595605011515634291, 11.23151124490415729773004093541, 12.17154957449262183513134288067, 13.96131331296397124215022305446, 15.13305897070850996033696219115, 15.92189822661643308350699551019

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