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Properties

Label 2-72-24.5-c4-0-3
Degree $2$
Conductor $72$
Sign $-0.0695 - 0.997i$
Analytic cond. $7.44263$
Root an. cond. $2.72811$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.93 − 0.723i)2-s + (14.9 + 5.69i)4-s − 10.6·5-s + 17.1·7-s + (−54.7 − 33.2i)8-s + (41.9 + 7.72i)10-s − 55.4·11-s + 95.6i·13-s + (−67.3 − 12.3i)14-s + (191. + 170. i)16-s + 365. i·17-s + 499. i·19-s + (−159. − 60.7i)20-s + (218. + 40.1i)22-s + 758. i·23-s + ⋯
L(s)  = 1  + (−0.983 − 0.180i)2-s + (0.934 + 0.355i)4-s − 0.426·5-s + 0.349·7-s + (−0.854 − 0.519i)8-s + (0.419 + 0.0772i)10-s − 0.458·11-s + 0.565i·13-s + (−0.343 − 0.0632i)14-s + (0.746 + 0.665i)16-s + 1.26i·17-s + 1.38i·19-s + (−0.398 − 0.151i)20-s + (0.450 + 0.0829i)22-s + 1.43i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0695 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0695 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.0695 - 0.997i$
Analytic conductor: \(7.44263\)
Root analytic conductor: \(2.72811\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :2),\ -0.0695 - 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.454782 + 0.487615i\)
\(L(\frac12)\) \(\approx\) \(0.454782 + 0.487615i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.93 + 0.723i)T \)
3 \( 1 \)
good5 \( 1 + 10.6T + 625T^{2} \)
7 \( 1 - 17.1T + 2.40e3T^{2} \)
11 \( 1 + 55.4T + 1.46e4T^{2} \)
13 \( 1 - 95.6iT - 2.85e4T^{2} \)
17 \( 1 - 365. iT - 8.35e4T^{2} \)
19 \( 1 - 499. iT - 1.30e5T^{2} \)
23 \( 1 - 758. iT - 2.79e5T^{2} \)
29 \( 1 - 958.T + 7.07e5T^{2} \)
31 \( 1 + 830.T + 9.23e5T^{2} \)
37 \( 1 + 2.03e3iT - 1.87e6T^{2} \)
41 \( 1 - 319. iT - 2.82e6T^{2} \)
43 \( 1 - 1.47e3iT - 3.41e6T^{2} \)
47 \( 1 - 847. iT - 4.87e6T^{2} \)
53 \( 1 - 3.83e3T + 7.89e6T^{2} \)
59 \( 1 + 3.44e3T + 1.21e7T^{2} \)
61 \( 1 + 4.80e3iT - 1.38e7T^{2} \)
67 \( 1 - 8.02e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.68e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.19e3T + 2.83e7T^{2} \)
79 \( 1 - 9.14e3T + 3.89e7T^{2} \)
83 \( 1 + 1.15e4T + 4.74e7T^{2} \)
89 \( 1 + 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 - 5.59e3T + 8.85e7T^{2} \)
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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

−14.37586632834540835056264408308, −12.75961472742197557825730296330, −11.74573769206442053503265411724, −10.76344211077597439582563417828, −9.667385487573285565511891208718, −8.331962714977032514705803637654, −7.51934786691763095332372552359, −5.94227223864484288835409365791, −3.74356556023582714697307525459, −1.72247726357328550555351166596, 0.47455251476122259676435173044, 2.67744118117683287128136618820, 5.04204205382780055414760194579, 6.75136019193750077726744367491, 7.85845322355295652362538349274, 8.884101086370450508316051111833, 10.18496224987269251485775657036, 11.19837409121303633188234729550, 12.17628267400534626346693459660, 13.72092545311905054318224240088

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