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Properties

Label 2-6e2-4.3-c32-0-47
Degree $2$
Conductor $36$
Sign $-0.522 + 0.852i$
Analytic cond. $233.519$
Root an. cond. $15.2813$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.20e4 − 5.71e4i)2-s + (−2.24e9 + 3.66e9i)4-s + 4.45e10·5-s + 4.42e13i·7-s + (2.81e14 + 1.12e13i)8-s + (−1.42e15 − 2.54e15i)10-s + 2.28e16i·11-s − 7.09e17·13-s + (2.53e18 − 1.41e18i)14-s + (−8.36e18 − 1.64e19i)16-s + 5.13e19·17-s − 9.08e19i·19-s + (−1.00e20 + 1.63e20i)20-s + (1.30e21 − 7.32e20i)22-s + 3.40e21i·23-s + ⋯
L(s)  = 1  + (−0.488 − 0.872i)2-s + (−0.522 + 0.852i)4-s + 0.292·5-s + 1.33i·7-s + (0.999 + 0.0398i)8-s + (−0.142 − 0.254i)10-s + 0.497i·11-s − 1.06·13-s + (1.16 − 0.650i)14-s + (−0.453 − 0.891i)16-s + 1.05·17-s − 0.315i·19-s + (−0.152 + 0.249i)20-s + (0.434 − 0.243i)22-s + 0.555i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(233.519\)
Root analytic conductor: \(15.2813\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :16),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.5342288161\)
\(L(\frac12)\) \(\approx\) \(0.5342288161\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.20e4 + 5.71e4i)T \)
3 \( 1 \)
good5 \( 1 - 4.45e10T + 2.32e22T^{2} \)
7 \( 1 - 4.42e13iT - 1.10e27T^{2} \)
11 \( 1 - 2.28e16iT - 2.11e33T^{2} \)
13 \( 1 + 7.09e17T + 4.42e35T^{2} \)
17 \( 1 - 5.13e19T + 2.36e39T^{2} \)
19 \( 1 + 9.08e19iT - 8.31e40T^{2} \)
23 \( 1 - 3.40e21iT - 3.76e43T^{2} \)
29 \( 1 - 9.75e22T + 6.26e46T^{2} \)
31 \( 1 + 8.90e23iT - 5.29e47T^{2} \)
37 \( 1 + 1.36e25T + 1.52e50T^{2} \)
41 \( 1 + 9.45e25T + 4.06e51T^{2} \)
43 \( 1 - 2.05e26iT - 1.86e52T^{2} \)
47 \( 1 + 6.70e25iT - 3.21e53T^{2} \)
53 \( 1 - 3.12e27T + 1.50e55T^{2} \)
59 \( 1 + 4.03e28iT - 4.64e56T^{2} \)
61 \( 1 + 4.61e28T + 1.35e57T^{2} \)
67 \( 1 - 2.83e29iT - 2.71e58T^{2} \)
71 \( 1 - 3.78e29iT - 1.73e59T^{2} \)
73 \( 1 + 8.28e29T + 4.22e59T^{2} \)
79 \( 1 + 3.22e30iT - 5.29e60T^{2} \)
83 \( 1 - 3.96e29iT - 2.57e61T^{2} \)
89 \( 1 + 1.44e31T + 2.40e62T^{2} \)
97 \( 1 - 5.69e30T + 3.77e63T^{2} \)
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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

−10.04269534707762972708170308432, −9.507808609388374272874966959458, −8.376268133820036394011002258755, −7.31666866475221409634163435839, −5.69462149815896543291574482335, −4.70538416544558397358921218091, −3.25819229116480846763569650729, −2.33853372742836858429389971553, −1.59161657041963512820517904123, −0.14350554819315498840121461918, 0.73421110655071246624286359290, 1.73102713860314203366565113120, 3.45019202114981505410669800370, 4.65999279062966783133540942493, 5.66764847668914506351738192690, 6.89743099887977459670555331796, 7.59831816124542536261542335978, 8.721369257217280494157508541386, 10.09247566473069438267840654134, 10.44530603891595672789696306363

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