Properties

Label 2-6e2-9.7-c5-0-4
Degree $2$
Conductor $36$
Sign $-0.227 + 0.973i$
Analytic cond. $5.77381$
Root an. cond. $2.40287$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.7 − 7.42i)3-s + (−55.1 − 95.6i)5-s + (−50.8 + 88.1i)7-s + (132. − 203. i)9-s + (75.1 − 130. i)11-s + (−317. − 550. i)13-s + (−1.46e3 − 900. i)15-s + 1.49e3·17-s + 1.43e3·19-s + (−43.3 + 1.58e3i)21-s + (632. + 1.09e3i)23-s + (−4.53e3 + 7.84e3i)25-s + (310. − 3.77e3i)27-s + (1.38e3 − 2.40e3i)29-s + (3.48e3 + 6.03e3i)31-s + ⋯
L(s)  = 1  + (0.879 − 0.476i)3-s + (−0.987 − 1.71i)5-s + (−0.392 + 0.679i)7-s + (0.546 − 0.837i)9-s + (0.187 − 0.324i)11-s + (−0.521 − 0.903i)13-s + (−1.68 − 1.03i)15-s + 1.25·17-s + 0.913·19-s + (−0.0214 + 0.784i)21-s + (0.249 + 0.431i)23-s + (−1.45 + 2.51i)25-s + (0.0819 − 0.996i)27-s + (0.306 − 0.531i)29-s + (0.651 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(5.77381\)
Root analytic conductor: \(2.40287\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.999078 - 1.25907i\)
\(L(\frac12)\) \(\approx\) \(0.999078 - 1.25907i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-13.7 + 7.42i)T \)
good5 \( 1 + (55.1 + 95.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (50.8 - 88.1i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-75.1 + 130. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (317. + 550. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.49e3T + 1.41e6T^{2} \)
19 \( 1 - 1.43e3T + 2.47e6T^{2} \)
23 \( 1 + (-632. - 1.09e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-1.38e3 + 2.40e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-3.48e3 - 6.03e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 7.95e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.01e3 - 1.75e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-6.26e3 + 1.08e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-3.24e3 + 5.61e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 9.82e3T + 4.18e8T^{2} \)
59 \( 1 + (2.35e4 + 4.07e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (4.16e3 - 7.22e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.63e3 - 6.28e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.58e3T + 1.80e9T^{2} \)
73 \( 1 - 5.80e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.18e4 + 5.52e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (4.14e4 - 7.17e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 3.86e3T + 5.58e9T^{2} \)
97 \( 1 + (3.46e4 - 5.99e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−15.31497505423430011093373244865, −13.74744633205761920733691631264, −12.43661676943908394402951224826, −12.11916277974677806929509352818, −9.576423513453576573167716666928, −8.538992958970472336249462351355, −7.63416236823180789926621708149, −5.29848874072152127915305679420, −3.40651888539169091238353049799, −0.925246792864447170007381711247, 2.92769571898548707814064235456, 4.06968519820545353222853560653, 6.95080623413675996937414437988, 7.75470690661575782180825708833, 9.710703224468444878610188346125, 10.63446281667951439177201512835, 11.96283605068126041250283511723, 13.93781169095464459665913856441, 14.54340955476286366010794674472, 15.52945059058818978200610805206

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