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Properties

Label 2-72-8.3-c10-0-44
Degree $2$
Conductor $72$
Sign $-0.228 - 0.973i$
Analytic cond. $45.7457$
Root an. cond. $6.76355$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−28.8 − 13.8i)2-s + (641. + 798. i)4-s − 5.38e3i·5-s − 6.61e3i·7-s + (−7.47e3 − 3.19e4i)8-s + (−7.44e4 + 1.55e5i)10-s − 2.21e5·11-s − 3.61e5i·13-s + (−9.14e4 + 1.90e5i)14-s + (−2.25e5 + 1.02e6i)16-s − 7.76e5·17-s + 1.46e6·19-s + (4.29e6 − 3.45e6i)20-s + (6.38e6 + 3.05e6i)22-s − 2.34e6i·23-s + ⋯
L(s)  = 1  + (−0.901 − 0.432i)2-s + (0.626 + 0.779i)4-s − 1.72i·5-s − 0.393i·7-s + (−0.228 − 0.973i)8-s + (−0.744 + 1.55i)10-s − 1.37·11-s − 0.974i·13-s + (−0.170 + 0.354i)14-s + (−0.215 + 0.976i)16-s − 0.546·17-s + 0.590·19-s + (1.34 − 1.07i)20-s + (1.23 + 0.593i)22-s − 0.364i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(45.7457\)
Root analytic conductor: \(6.76355\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :5),\ -0.228 - 0.973i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.256082 + 0.323009i\)
\(L(\frac12)\) \(\approx\) \(0.256082 + 0.323009i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (28.8 + 13.8i)T \)
3 \( 1 \)
good5 \( 1 + 5.38e3iT - 9.76e6T^{2} \)
7 \( 1 + 6.61e3iT - 2.82e8T^{2} \)
11 \( 1 + 2.21e5T + 2.59e10T^{2} \)
13 \( 1 + 3.61e5iT - 1.37e11T^{2} \)
17 \( 1 + 7.76e5T + 2.01e12T^{2} \)
19 \( 1 - 1.46e6T + 6.13e12T^{2} \)
23 \( 1 + 2.34e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.88e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.00e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.16e7iT - 4.80e15T^{2} \)
41 \( 1 + 4.70e7T + 1.34e16T^{2} \)
43 \( 1 - 3.52e7T + 2.16e16T^{2} \)
47 \( 1 - 4.23e8iT - 5.25e16T^{2} \)
53 \( 1 + 9.16e6iT - 1.74e17T^{2} \)
59 \( 1 - 1.03e9T + 5.11e17T^{2} \)
61 \( 1 - 1.37e8iT - 7.13e17T^{2} \)
67 \( 1 + 7.94e8T + 1.82e18T^{2} \)
71 \( 1 - 2.04e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.88e9T + 4.29e18T^{2} \)
79 \( 1 - 1.04e9iT - 9.46e18T^{2} \)
83 \( 1 + 3.77e9T + 1.55e19T^{2} \)
89 \( 1 - 4.07e9T + 3.11e19T^{2} \)
97 \( 1 - 1.76e9T + 7.37e19T^{2} \)
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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

−11.71997542576401298607457427884, −10.44384074570807117119309303578, −9.469688236119197809532031409334, −8.314635582593094602705080699152, −7.66097351386066990363103620294, −5.63912587047752330959544922874, −4.27871403695119197410898253442, −2.50253072650996179287484598223, −0.962948620420060941691241495175, −0.17862667799464532317815241976, 1.99232188615375620548883048547, 3.04592079497058397575870911286, 5.38411628331141772826720710834, 6.71357781162239258584912648001, 7.38838676857686086355768047738, 8.762557770910078657492810432125, 10.10963877947996195304673316051, 10.80387343947100054027739137600, 11.76922099371309688290478785033, 13.67961942589291899049222326910

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