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Properties

Label 2-72-8.5-c17-0-45
Degree $2$
Conductor $72$
Sign $0.692 + 0.721i$
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  + (257. − 254. i)2-s + (1.83e3 − 1.31e5i)4-s − 5.24e5i·5-s + 1.57e7·7-s + (−3.28e7 − 3.42e7i)8-s + (−1.33e8 − 1.35e8i)10-s + 2.44e8i·11-s + 2.66e9i·13-s + (4.06e9 − 4.00e9i)14-s + (−1.71e10 − 4.82e8i)16-s + 2.82e10·17-s + 7.79e10i·19-s + (−6.87e10 − 9.65e8i)20-s + (6.22e10 + 6.31e10i)22-s + 3.66e11·23-s + ⋯
L(s)  = 1  + (0.712 − 0.702i)2-s + (0.0140 − 0.999i)4-s − 0.600i·5-s + 1.03·7-s + (−0.692 − 0.721i)8-s + (−0.421 − 0.427i)10-s + 0.344i·11-s + 0.905i·13-s + (0.735 − 0.725i)14-s + (−0.999 − 0.0280i)16-s + 0.983·17-s + 1.05i·19-s + (−0.600 − 0.00843i)20-s + (0.241 + 0.245i)22-s + 0.976·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.692 + 0.721i$
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: \((2,\ 72,\ (\ :17/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(9)\) \(\approx\) \(4.113055580\)
\(L(\frac12)\) \(\approx\) \(4.113055580\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-257. + 254. i)T \)
3 \( 1 \)
good5 \( 1 + 5.24e5iT - 7.62e11T^{2} \)
7 \( 1 - 1.57e7T + 2.32e14T^{2} \)
11 \( 1 - 2.44e8iT - 5.05e17T^{2} \)
13 \( 1 - 2.66e9iT - 8.65e18T^{2} \)
17 \( 1 - 2.82e10T + 8.27e20T^{2} \)
19 \( 1 - 7.79e10iT - 5.48e21T^{2} \)
23 \( 1 - 3.66e11T + 1.41e23T^{2} \)
29 \( 1 - 2.78e11iT - 7.25e24T^{2} \)
31 \( 1 - 3.68e12T + 2.25e25T^{2} \)
37 \( 1 - 3.50e13iT - 4.56e26T^{2} \)
41 \( 1 - 3.67e13T + 2.61e27T^{2} \)
43 \( 1 - 1.25e14iT - 5.87e27T^{2} \)
47 \( 1 + 1.02e14T + 2.66e28T^{2} \)
53 \( 1 - 4.67e14iT - 2.05e29T^{2} \)
59 \( 1 - 4.08e13iT - 1.27e30T^{2} \)
61 \( 1 + 2.55e15iT - 2.24e30T^{2} \)
67 \( 1 - 2.46e15iT - 1.10e31T^{2} \)
71 \( 1 + 1.09e15T + 2.96e31T^{2} \)
73 \( 1 + 1.21e16T + 4.74e31T^{2} \)
79 \( 1 - 2.49e16T + 1.81e32T^{2} \)
83 \( 1 + 3.68e16iT - 4.21e32T^{2} \)
89 \( 1 - 4.15e16T + 1.37e33T^{2} \)
97 \( 1 - 7.13e15T + 5.95e33T^{2} \)
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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

−11.41291741182886746685477693288, −10.24007013562920802621681707889, −9.153728026911513574163759349543, −7.900146976354972357733346275089, −6.36382768879474841667690348189, −5.04075256681773139292501095087, −4.45792777771911041669502136445, −3.06657290211566761263467908564, −1.63371045071938971195579510012, −1.08164954650200606151428922684, 0.73253980753968800486124644345, 2.47916122929902043746876123869, 3.44414733203775536233404844722, 4.82188844660267271675849033851, 5.65714821529435817358108334530, 6.98029927841733338063373107688, 7.83819863793281079042988003505, 8.874841436357687456259007625080, 10.62656612118353730512225951729, 11.49069342664688102123598124998

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