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Properties

Label 2-72-8.5-c17-0-44
Degree $2$
Conductor $72$
Sign $0.755 - 0.655i$
Analytic cond. $131.919$
Root an. cond. $11.4856$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−351. + 85.4i)2-s + (1.16e5 − 6.01e4i)4-s + 4.65e5i·5-s + 2.24e7·7-s + (−3.58e7 + 3.10e7i)8-s + (−3.97e7 − 1.63e8i)10-s − 6.06e8i·11-s + 2.12e9i·13-s + (−7.90e9 + 1.92e9i)14-s + (9.95e9 − 1.40e10i)16-s + 5.45e9·17-s + 5.72e9i·19-s + (2.79e10 + 5.41e10i)20-s + (5.17e10 + 2.13e11i)22-s + 1.30e11·23-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s + (0.888 − 0.458i)4-s + 0.532i·5-s + 1.47·7-s + (−0.755 + 0.655i)8-s + (−0.125 − 0.517i)10-s − 0.852i·11-s + 0.723i·13-s + (−1.43 + 0.347i)14-s + (0.579 − 0.815i)16-s + 0.189·17-s + 0.0773i·19-s + (0.244 + 0.473i)20-s + (0.201 + 0.828i)22-s + 0.346·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(131.919\)
Root analytic conductor: \(11.4856\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :17/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.885171526\)
\(L(\frac12)\) \(\approx\) \(1.885171526\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (351. - 85.4i)T \)
3 \( 1 \)
good5 \( 1 - 4.65e5iT - 7.62e11T^{2} \)
7 \( 1 - 2.24e7T + 2.32e14T^{2} \)
11 \( 1 + 6.06e8iT - 5.05e17T^{2} \)
13 \( 1 - 2.12e9iT - 8.65e18T^{2} \)
17 \( 1 - 5.45e9T + 8.27e20T^{2} \)
19 \( 1 - 5.72e9iT - 5.48e21T^{2} \)
23 \( 1 - 1.30e11T + 1.41e23T^{2} \)
29 \( 1 - 4.23e11iT - 7.25e24T^{2} \)
31 \( 1 - 3.83e12T + 2.25e25T^{2} \)
37 \( 1 - 2.39e13iT - 4.56e26T^{2} \)
41 \( 1 - 5.79e12T + 2.61e27T^{2} \)
43 \( 1 + 4.00e13iT - 5.87e27T^{2} \)
47 \( 1 - 1.56e14T + 2.66e28T^{2} \)
53 \( 1 - 6.45e14iT - 2.05e29T^{2} \)
59 \( 1 + 8.37e14iT - 1.27e30T^{2} \)
61 \( 1 + 3.98e14iT - 2.24e30T^{2} \)
67 \( 1 + 5.22e15iT - 1.10e31T^{2} \)
71 \( 1 + 5.50e15T + 2.96e31T^{2} \)
73 \( 1 - 1.47e15T + 4.74e31T^{2} \)
79 \( 1 + 1.94e16T + 1.81e32T^{2} \)
83 \( 1 + 3.36e16iT - 4.21e32T^{2} \)
89 \( 1 - 1.09e16T + 1.37e33T^{2} \)
97 \( 1 - 6.57e16T + 5.95e33T^{2} \)
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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

−11.16976782130343356464199889297, −10.43384882974394639058902394141, −9.002407112184079034835113597862, −8.180552685243200616484796997628, −7.17880758212124885078382297151, −6.03786776678141136351036273329, −4.75492167815455723337602791568, −2.99507970370481238495365924584, −1.77902894225712814935627217220, −0.837970760351408803914963686425, 0.71263827929683162544643578825, 1.53503817696212840972764273154, 2.62716824392201465203662099737, 4.31245320734186143905875815093, 5.46944944318826252843064732860, 7.13194872411987197823810770008, 8.038628247510406808808922284536, 8.848886588439399780248450402954, 10.07440826278054170746469067794, 11.04389659389380983605531291948

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