Properties

Label 2-6e2-12.11-c15-0-12
Degree $2$
Conductor $36$
Sign $0.386 - 0.922i$
Analytic cond. $51.3696$
Root an. cond. $7.16726$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (179. − 19.7i)2-s + (3.19e4 − 7.11e3i)4-s + 2.20e5i·5-s − 6.10e5i·7-s + (5.61e6 − 1.91e6i)8-s + (4.35e6 + 3.96e7i)10-s − 4.61e7·11-s + 3.62e8·13-s + (−1.20e7 − 1.09e8i)14-s + (9.72e8 − 4.55e8i)16-s + 5.70e8i·17-s + 4.93e9i·19-s + (1.56e9 + 7.04e9i)20-s + (−8.30e9 + 9.12e8i)22-s − 1.55e10·23-s + ⋯
L(s)  = 1  + (0.994 − 0.109i)2-s + (0.976 − 0.217i)4-s + 1.26i·5-s − 0.280i·7-s + (0.946 − 0.322i)8-s + (0.137 + 1.25i)10-s − 0.713·11-s + 1.60·13-s + (−0.0306 − 0.278i)14-s + (0.905 − 0.423i)16-s + 0.337i·17-s + 1.26i·19-s + (0.273 + 1.23i)20-s + (−0.709 + 0.0779i)22-s − 0.950·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(51.3696\)
Root analytic conductor: \(7.16726\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :15/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(8)\) \(\approx\) \(4.111934757\)
\(L(\frac12)\) \(\approx\) \(4.111934757\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-179. + 19.7i)T \)
3 \( 1 \)
good5 \( 1 - 2.20e5iT - 3.05e10T^{2} \)
7 \( 1 + 6.10e5iT - 4.74e12T^{2} \)
11 \( 1 + 4.61e7T + 4.17e15T^{2} \)
13 \( 1 - 3.62e8T + 5.11e16T^{2} \)
17 \( 1 - 5.70e8iT - 2.86e18T^{2} \)
19 \( 1 - 4.93e9iT - 1.51e19T^{2} \)
23 \( 1 + 1.55e10T + 2.66e20T^{2} \)
29 \( 1 + 9.10e9iT - 8.62e21T^{2} \)
31 \( 1 - 3.03e11iT - 2.34e22T^{2} \)
37 \( 1 + 9.07e10T + 3.33e23T^{2} \)
41 \( 1 - 1.36e12iT - 1.55e24T^{2} \)
43 \( 1 + 3.26e11iT - 3.17e24T^{2} \)
47 \( 1 - 3.21e12T + 1.20e25T^{2} \)
53 \( 1 - 1.57e12iT - 7.31e25T^{2} \)
59 \( 1 + 3.62e13T + 3.65e26T^{2} \)
61 \( 1 - 1.89e13T + 6.02e26T^{2} \)
67 \( 1 - 4.38e13iT - 2.46e27T^{2} \)
71 \( 1 + 2.85e13T + 5.87e27T^{2} \)
73 \( 1 - 1.14e14T + 8.90e27T^{2} \)
79 \( 1 + 7.00e13iT - 2.91e28T^{2} \)
83 \( 1 - 2.99e14T + 6.11e28T^{2} \)
89 \( 1 - 5.14e14iT - 1.74e29T^{2} \)
97 \( 1 - 5.45e14T + 6.33e29T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.59273626347912693785974573017, −12.24708537920763046730263038494, −10.86433965383965386094054134865, −10.37688939250563796333966028733, −8.017163285808194667833313033305, −6.69763208975288126802638916495, −5.74451098182066823728212586179, −3.93380880665603734990591148413, −3.00129934486497339388323925881, −1.53811748507652028052971392437, 0.77248143078939451568231932390, 2.26548579782512670197707443283, 3.90917988271760015168505482178, 5.07832290931780590814492606324, 6.10578456528083758972741540840, 7.82768488679184102308189790372, 9.015998722086535191887024649410, 10.85135605314508765100470912226, 11.99807295071172079016715724162, 13.11006658089216507480387806488

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