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Properties

Label 2-72-24.11-c7-0-7
Degree $2$
Conductor $72$
Sign $-0.103 - 0.994i$
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  + (−3.16 + 10.8i)2-s + (−107. − 68.8i)4-s − 238.·5-s − 737. i·7-s + (1.08e3 − 954. i)8-s + (754. − 2.58e3i)10-s + 1.70e3i·11-s − 4.48e3i·13-s + (8.00e3 + 2.33e3i)14-s + (6.91e3 + 1.48e4i)16-s + 1.99e4i·17-s − 8.09e3·19-s + (2.57e4 + 1.63e4i)20-s + (−1.84e4 − 5.39e3i)22-s + 1.10e5·23-s + ⋯
L(s)  = 1  + (−0.280 + 0.959i)2-s + (−0.843 − 0.537i)4-s − 0.852·5-s − 0.812i·7-s + (0.752 − 0.658i)8-s + (0.238 − 0.818i)10-s + 0.385i·11-s − 0.565i·13-s + (0.780 + 0.227i)14-s + (0.421 + 0.906i)16-s + 0.985i·17-s − 0.270·19-s + (0.718 + 0.458i)20-s + (−0.370 − 0.107i)22-s + 1.89·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.103 - 0.994i$
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: \((2,\ 72,\ (\ :7/2),\ -0.103 - 0.994i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.685683 + 0.760846i\)
\(L(\frac12)\) \(\approx\) \(0.685683 + 0.760846i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.16 - 10.8i)T \)
3 \( 1 \)
good5 \( 1 + 238.T + 7.81e4T^{2} \)
7 \( 1 + 737. iT - 8.23e5T^{2} \)
11 \( 1 - 1.70e3iT - 1.94e7T^{2} \)
13 \( 1 + 4.48e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.99e4iT - 4.10e8T^{2} \)
19 \( 1 + 8.09e3T + 8.93e8T^{2} \)
23 \( 1 - 1.10e5T + 3.40e9T^{2} \)
29 \( 1 + 1.92e4T + 1.72e10T^{2} \)
31 \( 1 - 7.02e4iT - 2.75e10T^{2} \)
37 \( 1 - 4.41e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.82e5iT - 1.94e11T^{2} \)
43 \( 1 + 3.29e5T + 2.71e11T^{2} \)
47 \( 1 - 2.24e5T + 5.06e11T^{2} \)
53 \( 1 - 1.27e6T + 1.17e12T^{2} \)
59 \( 1 - 2.48e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.93e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.94e6T + 6.06e12T^{2} \)
71 \( 1 - 1.63e6T + 9.09e12T^{2} \)
73 \( 1 - 6.26e6T + 1.10e13T^{2} \)
79 \( 1 + 4.48e5iT - 1.92e13T^{2} \)
83 \( 1 - 3.63e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.41e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.10e7T + 8.07e13T^{2} \)
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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

−13.60289894872022983193534778885, −12.65098497163975452831267255794, −11.02691663669394192347179807792, −10.01424721756317497141208930691, −8.559939050651552384859820579597, −7.61850759651218124959536152489, −6.62498942771029495988816210863, −4.96392328343328207927725883261, −3.73535791608098521052542677419, −0.944461375530341234567882174284, 0.54519762938166855206765034967, 2.40962967738148514832259252835, 3.77118544509309640367741819630, 5.20112530892368260847222589104, 7.26647238350990038024748880029, 8.597298672492742499289572479148, 9.390163275140787418014248455678, 10.96727565084444593208511824421, 11.66923509682331504787113663423, 12.57388505223983068426955977127

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