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Properties

Label 2-72-24.11-c7-0-24
Degree $2$
Conductor $72$
Sign $0.999 - 0.0191i$
Analytic cond. $22.4917$
Root an. cond. $4.74254$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.7 + 3.47i)2-s + (103. + 74.7i)4-s + 527.·5-s − 1.31e3i·7-s + (858. + 1.16e3i)8-s + (5.68e3 + 1.83e3i)10-s − 961. i·11-s − 1.07e4i·13-s + (4.55e3 − 1.41e4i)14-s + (5.19e3 + 1.55e4i)16-s + 1.15e4i·17-s − 4.69e4·19-s + (5.48e4 + 3.94e4i)20-s + (3.33e3 − 1.03e4i)22-s + 8.87e3·23-s + ⋯
L(s)  = 1  + (0.951 + 0.307i)2-s + (0.811 + 0.584i)4-s + 1.88·5-s − 1.44i·7-s + (0.592 + 0.805i)8-s + (1.79 + 0.579i)10-s − 0.217i·11-s − 1.35i·13-s + (0.444 − 1.37i)14-s + (0.317 + 0.948i)16-s + 0.569i·17-s − 1.56·19-s + (1.53 + 1.10i)20-s + (0.0668 − 0.207i)22-s + 0.152·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0191i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.999 - 0.0191i$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ 0.999 - 0.0191i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.75798 + 0.0456625i\)
\(L(\frac12)\) \(\approx\) \(4.75798 + 0.0456625i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.7 - 3.47i)T \)
3 \( 1 \)
good5 \( 1 - 527.T + 7.81e4T^{2} \)
7 \( 1 + 1.31e3iT - 8.23e5T^{2} \)
11 \( 1 + 961. iT - 1.94e7T^{2} \)
13 \( 1 + 1.07e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.15e4iT - 4.10e8T^{2} \)
19 \( 1 + 4.69e4T + 8.93e8T^{2} \)
23 \( 1 - 8.87e3T + 3.40e9T^{2} \)
29 \( 1 - 1.09e5T + 1.72e10T^{2} \)
31 \( 1 - 2.39e5iT - 2.75e10T^{2} \)
37 \( 1 - 3.66e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.07e4iT - 1.94e11T^{2} \)
43 \( 1 - 2.25e5T + 2.71e11T^{2} \)
47 \( 1 + 7.13e5T + 5.06e11T^{2} \)
53 \( 1 + 7.86e5T + 1.17e12T^{2} \)
59 \( 1 - 1.19e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.03e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.72e6T + 6.06e12T^{2} \)
71 \( 1 - 8.27e4T + 9.09e12T^{2} \)
73 \( 1 + 2.92e6T + 1.10e13T^{2} \)
79 \( 1 + 1.21e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.39e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.64e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.75e5T + 8.07e13T^{2} \)
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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

−13.30640942740949789825110446077, −12.71048965500190163908433608185, −10.62019114015423439281353275193, −10.31984116516522308616195263510, −8.389601732939832476397661799647, −6.82627084019321434097020130677, −5.94991679691909746322150389837, −4.68655478569917530756997721594, −2.99721119061005414023387301611, −1.42541787283289010838869305812, 1.89057410818858744561032897032, 2.46867407435464532092012589424, 4.72119421894036216667057865513, 5.88501237665725381758219159485, 6.56006711715515787084599709474, 9.010588456384369246868191692765, 9.795334760187568439942240313602, 11.13266590390325488174518097087, 12.34724602468582957671043859659, 13.18423810681650430835973829025

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