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Properties

Label 2-6e2-36.31-c2-0-0
Degree $2$
Conductor $36$
Sign $-0.437 - 0.899i$
Analytic cond. $0.980928$
Root an. cond. $0.990418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 − 0.511i)2-s + (−2.76 + 1.16i)3-s + (3.47 + 1.97i)4-s + (−4.03 + 6.98i)5-s + (5.94 − 0.833i)6-s + (−3.90 + 2.25i)7-s + (−5.71 − 5.60i)8-s + (6.29 − 6.42i)9-s + (11.3 − 11.4i)10-s + (3.25 − 1.88i)11-s + (−11.9 − 1.42i)12-s + (−3.52 + 6.10i)13-s + (8.69 − 2.36i)14-s + (3.03 − 23.9i)15-s + (8.17 + 13.7i)16-s + 0.517·17-s + ⋯
L(s)  = 1  + (−0.966 − 0.255i)2-s + (−0.921 + 0.387i)3-s + (0.869 + 0.494i)4-s + (−0.806 + 1.39i)5-s + (0.990 − 0.138i)6-s + (−0.557 + 0.321i)7-s + (−0.713 − 0.700i)8-s + (0.699 − 0.714i)9-s + (1.13 − 1.14i)10-s + (0.296 − 0.171i)11-s + (−0.992 − 0.118i)12-s + (−0.271 + 0.469i)13-s + (0.621 − 0.168i)14-s + (0.202 − 1.59i)15-s + (0.511 + 0.859i)16-s + 0.0304·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.437 - 0.899i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ -0.437 - 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.193155 + 0.308889i\)
\(L(\frac12)\) \(\approx\) \(0.193155 + 0.308889i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.93 + 0.511i)T \)
3 \( 1 + (2.76 - 1.16i)T \)
good5 \( 1 + (4.03 - 6.98i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.90 - 2.25i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.25 + 1.88i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.52 - 6.10i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 0.517T + 289T^{2} \)
19 \( 1 - 16.4iT - 361T^{2} \)
23 \( 1 + (-27.7 - 15.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-9.48 - 16.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (13.1 + 7.58i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 0.592T + 1.36e3T^{2} \)
41 \( 1 + (-12.3 + 21.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-27.8 + 16.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (52.4 - 30.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 0.664T + 2.80e3T^{2} \)
59 \( 1 + (30.5 + 17.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-33.7 - 58.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (74.4 + 42.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + (-126. + 73.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-87.1 + 50.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 25.8T + 7.92e3T^{2} \)
97 \( 1 + (48.2 + 83.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−16.66300877267946472023770630335, −15.73334006615554944123672891297, −14.78008612002905710204531111783, −12.38739397524240124487894631085, −11.40223875030407305862138548038, −10.60613808830054553938771640559, −9.378234857204175858328963674874, −7.38140425355337377639464591233, −6.34738548189439628293495438021, −3.44418080994491580461684732215, 0.65066533261453962682350270084, 4.98744210778711723294875782044, 6.75386882050560075086661270665, 8.047472306510333086057737297663, 9.432835426798380861988215225238, 10.94605931908324960783939769802, 12.10380409269697131923618065609, 13.03025353183936961781014241005, 15.30431894650931013866800267117, 16.34894680876572511690866935861

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