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Properties

Label 2-6e2-4.3-c2-0-3
Degree $2$
Conductor $36$
Sign $0.5 + 0.866i$
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 − 1.73i)2-s + (−1.99 − 3.46i)4-s + 2·5-s + 6.92i·7-s − 7.99·8-s + (2 − 3.46i)10-s + 6.92i·11-s + 2·13-s + (11.9 + 6.92i)14-s + (−8 + 13.8i)16-s − 10·17-s − 20.7i·19-s + (−3.99 − 6.92i)20-s + (11.9 + 6.92i)22-s − 27.7i·23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.400·5-s + 0.989i·7-s − 0.999·8-s + (0.200 − 0.346i)10-s + 0.629i·11-s + 0.153·13-s + (0.857 + 0.494i)14-s + (−0.5 + 0.866i)16-s − 0.588·17-s − 1.09i·19-s + (−0.199 − 0.346i)20-s + (0.545 + 0.314i)22-s − 1.20i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08894 - 0.628699i\)
\(L(\frac12)\) \(\approx\) \(1.08894 - 0.628699i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 - 2T + 25T^{2} \)
7 \( 1 - 6.92iT - 49T^{2} \)
11 \( 1 - 6.92iT - 121T^{2} \)
13 \( 1 - 2T + 169T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 + 27.7iT - 529T^{2} \)
29 \( 1 - 26T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + 58T + 1.68e3T^{2} \)
43 \( 1 - 48.4iT - 1.84e3T^{2} \)
47 \( 1 - 69.2iT - 2.20e3T^{2} \)
53 \( 1 - 74T + 2.80e3T^{2} \)
59 \( 1 + 90.0iT - 3.48e3T^{2} \)
61 \( 1 - 26T + 3.72e3T^{2} \)
67 \( 1 + 6.92iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 - 48.4iT - 6.88e3T^{2} \)
89 \( 1 + 82T + 7.92e3T^{2} \)
97 \( 1 - 2T + 9.40e3T^{2} \)
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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

−15.69281492557054453653342039559, −14.74388229146239206075388990351, −13.45308087003564286452160478962, −12.43004054686086872016038275790, −11.31681003343307050804738487324, −9.916523791403926152757733473869, −8.774496596680792852849204754128, −6.28912218754223913325025705155, −4.73935364930487997163253497663, −2.44719771968788722276449401148, 3.84845937326100545215129282723, 5.68373852429476125954772253951, 7.10787343481272725408843793702, 8.489478930234701161067363574037, 10.12537533857628692118880300812, 11.80407577388763979492934445739, 13.42373903787653112614499162861, 13.88363228384763685657306666338, 15.28023555956808773268952114204, 16.46401896900918195063056214259

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