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Properties

Label 2-6e2-9.2-c4-0-2
Degree $2$
Conductor $36$
Sign $0.890 + 0.454i$
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  + (8.37 − 3.29i)3-s + (10.6 − 6.14i)5-s + (7.14 − 12.3i)7-s + (59.2 − 55.2i)9-s + (90.2 + 52.0i)11-s + (−37.6 − 65.1i)13-s + (68.8 − 86.4i)15-s + 341. i·17-s − 706.·19-s + (19.0 − 127. i)21-s + (−516. + 298. i)23-s + (−237. + 410. i)25-s + (314. − 657. i)27-s + (1.12e3 + 651. i)29-s + (−514. − 891. i)31-s + ⋯
L(s)  = 1  + (0.930 − 0.366i)3-s + (0.425 − 0.245i)5-s + (0.145 − 0.252i)7-s + (0.731 − 0.681i)9-s + (0.745 + 0.430i)11-s + (−0.222 − 0.385i)13-s + (0.305 − 0.384i)15-s + 1.18i·17-s − 1.95·19-s + (0.0431 − 0.288i)21-s + (−0.976 + 0.563i)23-s + (−0.379 + 0.657i)25-s + (0.430 − 0.902i)27-s + (1.34 + 0.774i)29-s + (−0.535 − 0.927i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.890 + 0.454i$
Analytic conductor: \(3.72131\)
Root analytic conductor: \(1.92907\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :2),\ 0.890 + 0.454i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.92038 - 0.461312i\)
\(L(\frac12)\) \(\approx\) \(1.92038 - 0.461312i\)
\(L(3)\) 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 \( 1 + (-8.37 + 3.29i)T \)
good5 \( 1 + (-10.6 + 6.14i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-7.14 + 12.3i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-90.2 - 52.0i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (37.6 + 65.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 341. iT - 8.35e4T^{2} \)
19 \( 1 + 706.T + 1.30e5T^{2} \)
23 \( 1 + (516. - 298. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.12e3 - 651. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (514. + 891. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 563.T + 1.87e6T^{2} \)
41 \( 1 + (-85.8 + 49.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-448. + 776. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-372. - 215. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 5.27e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.88e3 - 2.81e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (565. - 979. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-676. - 1.17e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.68e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.23e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.06e3 + 5.31e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-6.50e3 - 3.75e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 8.72e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.72e3 - 4.71e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.26778972136658858227996581793, −14.45583119966077294146768618022, −13.24584316883607711629261066380, −12.32553408713675104135658593172, −10.42054790481533771682845518342, −9.131514385257174893871888686222, −7.930624741979638347929228562602, −6.36329520297744281873853178816, −4.04923069743078285124672775145, −1.84721688687490204338019491049, 2.39211220861006147173067147021, 4.34915502316031118697124794171, 6.49835643224954538599673853024, 8.277012800430315066476544134471, 9.382317933207816750286565500629, 10.61564588524574775569682165033, 12.23353233653210797263190980427, 13.81747649075518718445670843756, 14.41810593087919850863494840408, 15.65905564058191198890295548607

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