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Properties

Label 2-6e2-9.7-c5-0-2
Degree $2$
Conductor $36$
Sign $0.108 + 0.994i$
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  + (−12.2 + 9.62i)3-s + (−14.0 − 24.3i)5-s + (75.7 − 131. i)7-s + (57.8 − 236. i)9-s + (138. − 240. i)11-s + (−291. − 505. i)13-s + (407. + 163. i)15-s − 1.61e3·17-s + 1.36e3·19-s + (333. + 2.33e3i)21-s + (−428. − 741. i)23-s + (1.16e3 − 2.02e3i)25-s + (1.56e3 + 3.45e3i)27-s + (−4.26e3 + 7.39e3i)29-s + (−1.46e3 − 2.54e3i)31-s + ⋯
L(s)  = 1  + (−0.786 + 0.617i)3-s + (−0.251 − 0.435i)5-s + (0.583 − 1.01i)7-s + (0.238 − 0.971i)9-s + (0.346 − 0.599i)11-s + (−0.479 − 0.829i)13-s + (0.467 + 0.187i)15-s − 1.35·17-s + 0.869·19-s + (0.164 + 1.15i)21-s + (−0.168 − 0.292i)23-s + (0.373 − 0.646i)25-s + (0.412 + 0.911i)27-s + (−0.942 + 1.63i)29-s + (−0.274 − 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.108 + 0.994i$
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: \((2,\ 36,\ (\ :5/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.696986 - 0.625051i\)
\(L(\frac12)\) \(\approx\) \(0.696986 - 0.625051i\)
\(L(\frac{7}{2})\) 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 \( 1 + (12.2 - 9.62i)T \)
good5 \( 1 + (14.0 + 24.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-75.7 + 131. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-138. + 240. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (291. + 505. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.61e3T + 1.41e6T^{2} \)
19 \( 1 - 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + (428. + 741. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (4.26e3 - 7.39e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (1.46e3 + 2.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 4.03e3T + 6.93e7T^{2} \)
41 \( 1 + (9.44e3 + 1.63e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-1.01e4 + 1.75e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-147. + 256. i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.03e3T + 4.18e8T^{2} \)
59 \( 1 + (-8.61e3 - 1.49e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.28e4 - 2.22e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.31e4 - 2.27e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.66e4T + 1.80e9T^{2} \)
73 \( 1 - 1.49e3T + 2.07e9T^{2} \)
79 \( 1 + (4.96e4 - 8.59e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.50e4 - 4.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + (-3.33e4 + 5.77e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−15.36837567297588219920101949888, −14.03740583238221792034318724595, −12.58037362207655876212564221989, −11.27805814222494630618484829898, −10.42367651389948802072662403989, −8.847819119115101233458536798449, −7.13551762706558436450904501546, −5.31591519936661240673279455411, −3.99534998016041612546391021968, −0.61280684167769166418064946995, 2.02586065988618080662748488835, 4.82476249176757688738235428179, 6.40872402117985270357826622893, 7.68079570728919165771519674565, 9.408832080248100072635627918830, 11.28696196984408255411580676379, 11.79728642845994429133629043182, 13.14819520722082835221845750840, 14.60716578014636527424232736109, 15.68533978810857972349864894534

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