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Properties

Label 2-72-8.5-c7-0-1
Degree $2$
Conductor $72$
Sign $-0.874 + 0.484i$
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  + (−11.1 + 1.89i)2-s + (120. − 42.3i)4-s + 338. i·5-s − 438.·7-s + (−1.26e3 + 701. i)8-s + (−642. − 3.77e3i)10-s + 1.96e3i·11-s + 2.21e3i·13-s + (4.89e3 − 833. i)14-s + (1.27e4 − 1.02e4i)16-s + 1.21e4·17-s + 3.28e4i·19-s + (1.43e4 + 4.08e4i)20-s + (−3.73e3 − 2.19e4i)22-s − 1.96e4·23-s + ⋯
L(s)  = 1  + (−0.985 + 0.167i)2-s + (0.943 − 0.330i)4-s + 1.21i·5-s − 0.483·7-s + (−0.874 + 0.484i)8-s + (−0.203 − 1.19i)10-s + 0.445i·11-s + 0.279i·13-s + (0.476 − 0.0811i)14-s + (0.781 − 0.624i)16-s + 0.598·17-s + 1.09i·19-s + (0.400 + 1.14i)20-s + (−0.0747 − 0.439i)22-s − 0.335·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.874 + 0.484i$
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: \((2,\ 72,\ (\ :7/2),\ -0.874 + 0.484i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0658226 - 0.254754i\)
\(L(\frac12)\) \(\approx\) \(0.0658226 - 0.254754i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.1 - 1.89i)T \)
3 \( 1 \)
good5 \( 1 - 338. iT - 7.81e4T^{2} \)
7 \( 1 + 438.T + 8.23e5T^{2} \)
11 \( 1 - 1.96e3iT - 1.94e7T^{2} \)
13 \( 1 - 2.21e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.21e4T + 4.10e8T^{2} \)
19 \( 1 - 3.28e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.96e4T + 3.40e9T^{2} \)
29 \( 1 + 1.60e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.29e5T + 2.75e10T^{2} \)
37 \( 1 + 4.96e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.99e5T + 1.94e11T^{2} \)
43 \( 1 - 8.83e4iT - 2.71e11T^{2} \)
47 \( 1 + 8.20e5T + 5.06e11T^{2} \)
53 \( 1 - 1.53e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.82e6iT - 2.48e12T^{2} \)
61 \( 1 + 4.84e5iT - 3.14e12T^{2} \)
67 \( 1 + 7.98e4iT - 6.06e12T^{2} \)
71 \( 1 + 1.27e6T + 9.09e12T^{2} \)
73 \( 1 - 3.70e6T + 1.10e13T^{2} \)
79 \( 1 + 2.55e6T + 1.92e13T^{2} \)
83 \( 1 + 1.53e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.99e6T + 4.42e13T^{2} \)
97 \( 1 + 2.89e4T + 8.07e13T^{2} \)
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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

−14.21587100772732369902184546830, −12.46384923180672771863004911874, −11.29612695150281477225147519187, −10.28429610829567998813756365991, −9.508394072196889773100326914770, −7.927431290486798793049971239412, −6.94556965670812110472321841823, −5.90527186896910653998969792303, −3.40715880262284872740108525652, −1.95501909187412451312811374304, 0.12683193950122045422149341679, 1.38322821644698460879146032104, 3.25508714501174123310737236140, 5.23970764719000336493156881554, 6.78198619768351734910138602511, 8.220493579278695165927205597653, 9.035154969557980494528495601913, 10.05907481417222570861195057053, 11.36116936633452887914458142122, 12.45182665742906201604716346386

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