↑

Properties

Label 2-72-9.5-c2-0-0
Degree $2$
Conductor $72$
Sign $0.766 - 0.642i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + (6.39 + 3.69i)5-s + (3.39 + 5.88i)7-s + 9·9-s + (5.29 − 3.05i)11-s + (−8.39 + 14.5i)13-s + (−19.1 − 11.0i)15-s − 25.1i·17-s − 17.5·19-s + (−10.1 − 17.6i)21-s + (12.3 + 7.15i)23-s + (14.7 + 25.6i)25-s − 27·27-s + (16.1 − 9.35i)29-s + (23.3 − 40.5i)31-s + ⋯
L(s)  = 1  − 3-s + (1.27 + 0.738i)5-s + (0.485 + 0.841i)7-s + 9-s + (0.481 − 0.278i)11-s + (−0.646 + 1.11i)13-s + (−1.27 − 0.738i)15-s − 1.48i·17-s − 0.926·19-s + (−0.485 − 0.841i)21-s + (0.539 + 0.311i)23-s + (0.591 + 1.02i)25-s − 27-s + (0.558 − 0.322i)29-s + (0.754 − 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07520 + 0.391341i\)
\(L(\frac12)\) \(\approx\) \(1.07520 + 0.391341i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
good5 \( 1 + (-6.39 - 3.69i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.39 - 5.88i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.29 + 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.39 - 14.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 25.1iT - 289T^{2} \)
19 \( 1 + 17.5T + 361T^{2} \)
23 \( 1 + (-12.3 - 7.15i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.1 + 9.35i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23.3 + 40.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 49.5T + 1.36e3T^{2} \)
41 \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-22.0 - 38.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.8 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 10.1iT - 2.80e3T^{2} \)
59 \( 1 + (14.2 + 8.25i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.6 + 18.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.4 + 75.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 30.2iT - 5.04e3T^{2} \)
73 \( 1 + 48.7T + 5.32e3T^{2} \)
79 \( 1 + (55.7 + 96.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (85.0 - 49.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 75.5iT - 7.92e3T^{2} \)
97 \( 1 + (-70.2 - 121. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−14.41688325258846776342743691277, −13.53006003149772752112338470190, −12.04161191162439038463658328682, −11.32046801022467912025839057560, −10.05830690372472960172821180596, −9.122837994284922126541428454773, −6.99753850693297386529697049189, −6.07671539573284741144079914708, −4.87480411787389668835497170566, −2.17816287193215497714178442333, 1.39329167615973852944354652640, 4.53777492915399931769010100391, 5.64066693018669042413509556559, 6.87936696258706295113441239942, 8.572971122942973807699475946149, 10.18390852892272240559590871445, 10.58535982153763876250174509171, 12.34028740130042524784887986393, 12.94594623415847350675080962230, 14.13474884855120098574955395150

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