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Properties

Label 2-72-8.5-c3-0-3
Degree $2$
Conductor $72$
Sign $-0.883 - 0.467i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 2.64i)2-s + (−6.00 + 5.29i)4-s + 10.5i·5-s − 8·7-s + (−20.0 − 10.5i)8-s + (−28.0 + 10.5i)10-s + 15.8i·11-s + 52.9i·13-s + (−8 − 21.1i)14-s + (8.00 − 63.4i)16-s + 14·17-s − 37.0i·19-s + (−56.0 − 63.4i)20-s + (−42.0 + 15.8i)22-s + 152·23-s + ⋯
L(s)  = 1  + (0.353 + 0.935i)2-s + (−0.750 + 0.661i)4-s + 0.946i·5-s − 0.431·7-s + (−0.883 − 0.467i)8-s + (−0.885 + 0.334i)10-s + 0.435i·11-s + 1.12i·13-s + (−0.152 − 0.404i)14-s + (0.125 − 0.992i)16-s + 0.199·17-s − 0.447i·19-s + (−0.626 − 0.709i)20-s + (−0.407 + 0.153i)22-s + 1.37·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.883 - 0.467i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.330597 + 1.33161i\)
\(L(\frac12)\) \(\approx\) \(0.330597 + 1.33161i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 2.64i)T \)
3 \( 1 \)
good5 \( 1 - 10.5iT - 125T^{2} \)
7 \( 1 + 8T + 343T^{2} \)
11 \( 1 - 15.8iT - 1.33e3T^{2} \)
13 \( 1 - 52.9iT - 2.19e3T^{2} \)
17 \( 1 - 14T + 4.91e3T^{2} \)
19 \( 1 + 37.0iT - 6.85e3T^{2} \)
23 \( 1 - 152T + 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 - 224T + 2.97e4T^{2} \)
37 \( 1 - 243. iT - 5.06e4T^{2} \)
41 \( 1 - 70T + 6.89e4T^{2} \)
43 \( 1 + 439. iT - 7.95e4T^{2} \)
47 \( 1 + 336T + 1.03e5T^{2} \)
53 \( 1 + 31.7iT - 1.48e5T^{2} \)
59 \( 1 + 534. iT - 2.05e5T^{2} \)
61 \( 1 - 95.2iT - 2.26e5T^{2} \)
67 \( 1 - 174. iT - 3.00e5T^{2} \)
71 \( 1 - 72T + 3.57e5T^{2} \)
73 \( 1 + 294T + 3.89e5T^{2} \)
79 \( 1 + 464T + 4.93e5T^{2} \)
83 \( 1 - 545. iT - 5.71e5T^{2} \)
89 \( 1 + 266T + 7.04e5T^{2} \)
97 \( 1 - 994T + 9.12e5T^{2} \)
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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

−14.65963823743256225283344443442, −13.78702077687976888808065681067, −12.67656760022398370189261486050, −11.40677292306453169124835243864, −9.917028316171198649075968263845, −8.731674331304075675191967026619, −7.11872637359071948440203782159, −6.54096590225563299800493585797, −4.80451002701726398848966697391, −3.15194866284524281273984399844, 0.854896124957146314057118261759, 3.08563278819397497509825045457, 4.72665515713382972390616266655, 5.95039342288443321110872438606, 8.171957454114703233980480392527, 9.297023468375512313024728390392, 10.37181157906699193846962606930, 11.58641364709414632187498661563, 12.76648181828554552873885213354, 13.19866365658306888101224957965

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