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Properties

Label 2-6e2-36.23-c3-0-9
Degree $2$
Conductor $36$
Sign $0.932 + 0.360i$
Analytic cond. $2.12406$
Root an. cond. $1.45741$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 − 1.83i)2-s + (2.90 + 4.30i)3-s + (1.26 − 7.89i)4-s + (2.08 + 1.20i)5-s + (14.1 + 3.94i)6-s + (−2.30 + 1.32i)7-s + (−11.7 − 19.3i)8-s + (−10.1 + 25.0i)9-s + (6.70 − 1.23i)10-s + (−24.1 − 41.8i)11-s + (37.7 − 17.4i)12-s + (−20.3 + 35.2i)13-s + (−2.51 + 7.09i)14-s + (0.870 + 12.4i)15-s + (−60.8 − 19.9i)16-s + 36.3i·17-s + ⋯
L(s)  = 1  + (0.760 − 0.648i)2-s + (0.559 + 0.829i)3-s + (0.158 − 0.987i)4-s + (0.186 + 0.107i)5-s + (0.963 + 0.268i)6-s + (−0.124 + 0.0718i)7-s + (−0.520 − 0.853i)8-s + (−0.374 + 0.927i)9-s + (0.211 − 0.0391i)10-s + (−0.661 − 1.14i)11-s + (0.907 − 0.420i)12-s + (−0.434 + 0.752i)13-s + (−0.0480 + 0.135i)14-s + (0.0149 + 0.214i)15-s + (−0.950 − 0.312i)16-s + 0.518i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96787 - 0.366786i\)
\(L(\frac12)\) \(\approx\) \(1.96787 - 0.366786i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.15 + 1.83i)T \)
3 \( 1 + (-2.90 - 4.30i)T \)
good5 \( 1 + (-2.08 - 1.20i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (2.30 - 1.32i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (24.1 + 41.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.3 - 35.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 36.3iT - 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 + (-97.0 + 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-153. + 88.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-152. - 87.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 + (201. + 116. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-252. + 145. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-30.9 - 53.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 352. iT - 1.48e5T^{2} \)
59 \( 1 + (70.7 - 122. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (7.71 + 13.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (131. + 76.0i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 28.0T + 3.57e5T^{2} \)
73 \( 1 - 124.T + 3.89e5T^{2} \)
79 \( 1 + (648. - 374. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-174. - 302. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 416. iT - 7.04e5T^{2} \)
97 \( 1 + (752. + 1.30e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.67646117474313705801435196362, −14.42729505954050157570760162005, −13.78348582909995472318963416967, −12.31899196655105760235134661700, −10.82598942583275031289239312696, −9.991199684487625784647706058264, −8.474353758773043627258499486610, −6.01661162307282755849278859840, −4.38010797624303035068301594946, −2.74913381958337871818210976664, 2.81428427048082872587695760096, 5.10157510014861269944579095865, 6.90624169677435803286941895823, 7.81081631706341863359211243106, 9.417205217031215181157401570283, 11.65871819199298385098871112172, 12.96339075445117109533720585247, 13.44363305785012762095933787336, 14.90356941154229789708595420015, 15.59443653907072269529976656756

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