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Properties

Label 2-72-8.5-c17-0-4
Degree $2$
Conductor $72$
Sign $-0.946 + 0.321i$
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  + (−214. − 291. i)2-s + (−3.93e4 + 1.25e5i)4-s + 1.21e6i·5-s + 1.76e7·7-s + (4.49e7 − 1.52e7i)8-s + (3.53e8 − 2.59e8i)10-s + 7.26e8i·11-s + 3.16e9i·13-s + (−3.78e9 − 5.15e9i)14-s + (−1.40e10 − 9.84e9i)16-s − 4.56e10·17-s + 1.55e10i·19-s + (−1.51e11 − 4.77e10i)20-s + (2.12e11 − 1.55e11i)22-s − 4.09e11·23-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.300 + 0.953i)4-s + 1.38i·5-s + 1.15·7-s + (0.946 − 0.321i)8-s + (1.11 − 0.821i)10-s + 1.02i·11-s + 1.07i·13-s + (−0.684 − 0.933i)14-s + (−0.819 − 0.573i)16-s − 1.58·17-s + 0.210i·19-s + (−1.32 − 0.417i)20-s + (0.824 − 0.604i)22-s − 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.946 + 0.321i$
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: \((2,\ 72,\ (\ :17/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.4594832798\)
\(L(\frac12)\) \(\approx\) \(0.4594832798\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (214. + 291. i)T \)
3 \( 1 \)
good5 \( 1 - 1.21e6iT - 7.62e11T^{2} \)
7 \( 1 - 1.76e7T + 2.32e14T^{2} \)
11 \( 1 - 7.26e8iT - 5.05e17T^{2} \)
13 \( 1 - 3.16e9iT - 8.65e18T^{2} \)
17 \( 1 + 4.56e10T + 8.27e20T^{2} \)
19 \( 1 - 1.55e10iT - 5.48e21T^{2} \)
23 \( 1 + 4.09e11T + 1.41e23T^{2} \)
29 \( 1 - 3.11e12iT - 7.25e24T^{2} \)
31 \( 1 + 6.57e12T + 2.25e25T^{2} \)
37 \( 1 + 1.36e13iT - 4.56e26T^{2} \)
41 \( 1 - 2.81e12T + 2.61e27T^{2} \)
43 \( 1 - 1.36e13iT - 5.87e27T^{2} \)
47 \( 1 + 6.84e11T + 2.66e28T^{2} \)
53 \( 1 + 2.47e14iT - 2.05e29T^{2} \)
59 \( 1 + 2.03e15iT - 1.27e30T^{2} \)
61 \( 1 + 8.92e14iT - 2.24e30T^{2} \)
67 \( 1 - 5.06e15iT - 1.10e31T^{2} \)
71 \( 1 - 6.85e15T + 2.96e31T^{2} \)
73 \( 1 - 1.37e15T + 4.74e31T^{2} \)
79 \( 1 + 1.78e16T + 1.81e32T^{2} \)
83 \( 1 - 1.48e15iT - 4.21e32T^{2} \)
89 \( 1 - 3.78e16T + 1.37e33T^{2} \)
97 \( 1 - 1.28e17T + 5.95e33T^{2} \)
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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

−11.37925987956447392038408542725, −11.00829762419632117185244681932, −9.891652348871037966950032286994, −8.748934319186691107351546247114, −7.48436374097061072753528468694, −6.73752626261237048333181942166, −4.69836702071124441247276340396, −3.70745087601202091925815332661, −2.15013265165147768600678841397, −1.84709857836385458158522349735, 0.12757972351518065305483993686, 0.914864646692162502157901415102, 1.99977915863168570770541689381, 4.26669801049505279933034640029, 5.15914194187501974059821014828, 6.03999232166301845182056044114, 7.73554549436890878738500374043, 8.433237095394637265063424553193, 9.118836216862253492063319812734, 10.57991206932342195182129182558

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