Properties

Label 2-72-8.5-c7-0-18
Degree $2$
Conductor $72$
Sign $0.401 - 0.915i$
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  + (−1.55 + 11.2i)2-s + (−123. − 34.7i)4-s + 184. i·5-s + 1.05e3·7-s + (581. − 1.32e3i)8-s + (−2.07e3 − 286. i)10-s − 4.32e3i·11-s − 1.12e4i·13-s + (−1.63e3 + 1.17e4i)14-s + (1.39e4 + 8.57e3i)16-s + 2.17e4·17-s + 4.54e4i·19-s + (6.43e3 − 2.27e4i)20-s + (4.84e4 + 6.71e3i)22-s − 4.41e3·23-s + ⋯
L(s)  = 1  + (−0.137 + 0.990i)2-s + (−0.962 − 0.271i)4-s + 0.661i·5-s + 1.15·7-s + (0.401 − 0.915i)8-s + (−0.655 − 0.0907i)10-s − 0.979i·11-s − 1.42i·13-s + (−0.159 + 1.14i)14-s + (0.852 + 0.523i)16-s + 1.07·17-s + 1.52i·19-s + (0.179 − 0.636i)20-s + (0.970 + 0.134i)22-s − 0.0756·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.401 - 0.915i$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ 0.401 - 0.915i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.57540 + 1.02985i\)
\(L(\frac12)\) \(\approx\) \(1.57540 + 1.02985i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.55 - 11.2i)T \)
3 \( 1 \)
good5 \( 1 - 184. iT - 7.81e4T^{2} \)
7 \( 1 - 1.05e3T + 8.23e5T^{2} \)
11 \( 1 + 4.32e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.12e4iT - 6.27e7T^{2} \)
17 \( 1 - 2.17e4T + 4.10e8T^{2} \)
19 \( 1 - 4.54e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.41e3T + 3.40e9T^{2} \)
29 \( 1 + 2.36e4iT - 1.72e10T^{2} \)
31 \( 1 - 7.29e4T + 2.75e10T^{2} \)
37 \( 1 - 4.83e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.11e5T + 1.94e11T^{2} \)
43 \( 1 + 9.61e4iT - 2.71e11T^{2} \)
47 \( 1 - 1.56e5T + 5.06e11T^{2} \)
53 \( 1 - 6.86e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.79e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.36e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.08e6iT - 6.06e12T^{2} \)
71 \( 1 - 5.60e6T + 9.09e12T^{2} \)
73 \( 1 - 2.16e4T + 1.10e13T^{2} \)
79 \( 1 - 2.34e6T + 1.92e13T^{2} \)
83 \( 1 - 8.82e5iT - 2.71e13T^{2} \)
89 \( 1 - 1.34e6T + 4.42e13T^{2} \)
97 \( 1 - 7.32e6T + 8.07e13T^{2} \)
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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

−13.87882967318711790279524733239, −12.44044130671558826594238449805, −10.92642786077765605548408682702, −10.00133806754362990863205833154, −8.249270240551313115214571005030, −7.81006537258336803148200002286, −6.16111213379166942824755909798, −5.16598520604609426723549923075, −3.40393482504038123964514817397, −0.992812783638979849465307777037, 1.03285126054594715312105359458, 2.21177040477814009356318294427, 4.27880340255862778488222778604, 5.06772755706755687946454706991, 7.38673899764155995662081544882, 8.701866559018465577186058769091, 9.544647595020807523268226201631, 10.94806116737134192763597277564, 11.83806815609742576917068422945, 12.70512902151627886283636786399

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