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Properties

Label 2-72-8.5-c7-0-28
Degree $2$
Conductor $72$
Sign $-0.506 + 0.862i$
Analytic cond. $22.4917$
Root an. cond. $4.74254$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.6 − 3.84i)2-s + (98.4 − 81.8i)4-s + 209. i·5-s − 1.30e3·7-s + (732. − 1.24e3i)8-s + (805. + 2.22e3i)10-s − 7.07e3i·11-s − 1.43e4i·13-s + (−1.38e4 + 5.00e3i)14-s + (2.99e3 − 1.61e4i)16-s + 1.91e4·17-s + 9.61e3i·19-s + (1.71e4 + 2.06e4i)20-s + (−2.71e4 − 7.52e4i)22-s − 8.66e4·23-s + ⋯
L(s)  = 1  + (0.940 − 0.339i)2-s + (0.769 − 0.639i)4-s + 0.749i·5-s − 1.43·7-s + (0.506 − 0.862i)8-s + (0.254 + 0.704i)10-s − 1.60i·11-s − 1.80i·13-s + (−1.34 + 0.487i)14-s + (0.182 − 0.983i)16-s + 0.947·17-s + 0.321i·19-s + (0.479 + 0.576i)20-s + (−0.544 − 1.50i)22-s − 1.48·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.506 + 0.862i$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ -0.506 + 0.862i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.15684 - 2.02012i\)
\(L(\frac12)\) \(\approx\) \(1.15684 - 2.02012i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.6 + 3.84i)T \)
3 \( 1 \)
good5 \( 1 - 209. iT - 7.81e4T^{2} \)
7 \( 1 + 1.30e3T + 8.23e5T^{2} \)
11 \( 1 + 7.07e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.43e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.91e4T + 4.10e8T^{2} \)
19 \( 1 - 9.61e3iT - 8.93e8T^{2} \)
23 \( 1 + 8.66e4T + 3.40e9T^{2} \)
29 \( 1 + 1.01e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.84e4T + 2.75e10T^{2} \)
37 \( 1 - 7.62e4iT - 9.49e10T^{2} \)
41 \( 1 + 5.10e5T + 1.94e11T^{2} \)
43 \( 1 - 7.84e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.80e5T + 5.06e11T^{2} \)
53 \( 1 + 8.80e5iT - 1.17e12T^{2} \)
59 \( 1 - 9.25e4iT - 2.48e12T^{2} \)
61 \( 1 - 2.40e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.99e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.60e6T + 9.09e12T^{2} \)
73 \( 1 - 1.68e6T + 1.10e13T^{2} \)
79 \( 1 - 4.81e6T + 1.92e13T^{2} \)
83 \( 1 - 1.23e6iT - 2.71e13T^{2} \)
89 \( 1 - 9.48e6T + 4.42e13T^{2} \)
97 \( 1 - 1.45e7T + 8.07e13T^{2} \)
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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

−12.96210064533031692800801178171, −11.87629754612749672623532288944, −10.56356214366950141053382728012, −9.963276361516382872937873607585, −7.931225306284777321426214584572, −6.34178212551733600168979988172, −5.70779482214176237728272195930, −3.45301054382427010475052078927, −2.96908294845763797888259816060, −0.56104401571099473706054394374, 1.98477752510057285804255787090, 3.75870997744581607436282391542, 4.88132679910924757721398529081, 6.39999864190110334473553566563, 7.29191017011495685546559662938, 9.022287026506849408097901384589, 10.10549327486582765647792207935, 12.04593512731777865747087126215, 12.40034271729538082458843643548, 13.50458256833282165654570690847

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