↑

Properties

Label 2-72-8.3-c6-0-7
Degree $2$
Conductor $72$
Sign $-0.961 + 0.274i$
Analytic cond. $16.5638$
Root an. cond. $4.06987$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.62 + 6.52i)2-s + (−21.2 + 60.3i)4-s + 199. i·5-s + 19.6i·7-s + (−492. + 140. i)8-s + (−1.29e3 + 920. i)10-s + 924.·11-s − 1.55e3i·13-s + (−128. + 90.9i)14-s + (−3.19e3 − 2.56e3i)16-s − 5.14e3·17-s − 1.69e3·19-s + (−1.20e4 − 4.22e3i)20-s + (4.27e3 + 6.03e3i)22-s + 1.92e4i·23-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)2-s + (−0.331 + 0.943i)4-s + 1.59i·5-s + 0.0573i·7-s + (−0.961 + 0.274i)8-s + (−1.29 + 0.920i)10-s + 0.694·11-s − 0.705i·13-s + (−0.0467 + 0.0331i)14-s + (−0.779 − 0.626i)16-s − 1.04·17-s − 0.247·19-s + (−1.50 − 0.528i)20-s + (0.401 + 0.566i)22-s + 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.247503 - 1.77047i\)
\(L(\frac12)\) \(\approx\) \(0.247503 - 1.77047i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.62 - 6.52i)T \)
3 \( 1 \)
good5 \( 1 - 199. iT - 1.56e4T^{2} \)
7 \( 1 - 19.6iT - 1.17e5T^{2} \)
11 \( 1 - 924.T + 1.77e6T^{2} \)
13 \( 1 + 1.55e3iT - 4.82e6T^{2} \)
17 \( 1 + 5.14e3T + 2.41e7T^{2} \)
19 \( 1 + 1.69e3T + 4.70e7T^{2} \)
23 \( 1 - 1.92e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.65e4iT - 5.94e8T^{2} \)
31 \( 1 + 7.55e3iT - 8.87e8T^{2} \)
37 \( 1 - 2.89e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.21e4T + 4.75e9T^{2} \)
43 \( 1 - 5.89e3T + 6.32e9T^{2} \)
47 \( 1 - 6.44e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.97e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.42e5T + 4.21e10T^{2} \)
61 \( 1 - 9.64e4iT - 5.15e10T^{2} \)
67 \( 1 + 7.52e4T + 9.04e10T^{2} \)
71 \( 1 - 5.56e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.85e5T + 1.51e11T^{2} \)
79 \( 1 - 3.42e5iT - 2.43e11T^{2} \)
83 \( 1 - 9.29e5T + 3.26e11T^{2} \)
89 \( 1 + 4.34e5T + 4.96e11T^{2} \)
97 \( 1 - 6.43e5T + 8.32e11T^{2} \)
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

−14.13935695823119538967533091396, −13.30109067079019141852999346549, −11.84061982432422918253350881331, −10.85425781004726167645543776277, −9.362654962318828889631596875615, −7.79802744147103975103682721718, −6.81476543583410742597067926383, −5.84913819229800943728098728800, −3.99391701855247016380248175575, −2.72862168625655847689505665458, 0.56600724290255138206839217556, 1.94056171825976796535114757984, 4.09466344967947526616364491680, 4.93325944306615519389551693564, 6.45269365035294281883050051230, 8.685417759548323785431873357980, 9.284244917000013680981393622307, 10.78331105196332676155409406489, 11.99805204925575110291912820572, 12.68198291405436462035763807518

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