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Properties

Label 2-72-8.5-c7-0-32
Degree $2$
Conductor $72$
Sign $-i$
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  − 11.3i·2-s − 128.·4-s − 557. i·5-s − 754·7-s + 1.44e3i·8-s − 6.30e3·10-s − 6.72e3i·11-s + 8.53e3i·14-s + 1.63e4·16-s + 7.13e4i·20-s − 7.60e4·22-s − 2.32e5·25-s + 9.65e4·28-s + 2.51e5i·29-s + 3.31e5·31-s − 1.85e5i·32-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 1.99i·5-s − 0.830·7-s + 1.00i·8-s − 1.99·10-s − 1.52i·11-s + 0.830i·14-s + 1.00·16-s + 1.99i·20-s − 1.52·22-s − 2.97·25-s + 0.830·28-s + 1.91i·29-s + 1.99·31-s − 1.00i·32-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(4)\) \(\approx\) \(0.511711 + 0.511711i\)
\(L(\frac12)\) \(\approx\) \(0.511711 + 0.511711i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 11.3iT \)
3 \( 1 \)
good5 \( 1 + 557. iT - 7.81e4T^{2} \)
7 \( 1 + 754T + 8.23e5T^{2} \)
11 \( 1 + 6.72e3iT - 1.94e7T^{2} \)
13 \( 1 - 6.27e7T^{2} \)
17 \( 1 + 4.10e8T^{2} \)
19 \( 1 - 8.93e8T^{2} \)
23 \( 1 + 3.40e9T^{2} \)
29 \( 1 - 2.51e5iT - 1.72e10T^{2} \)
31 \( 1 - 3.31e5T + 2.75e10T^{2} \)
37 \( 1 - 9.49e10T^{2} \)
41 \( 1 + 1.94e11T^{2} \)
43 \( 1 - 2.71e11T^{2} \)
47 \( 1 + 5.06e11T^{2} \)
53 \( 1 + 2.39e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.48e6iT - 2.48e12T^{2} \)
61 \( 1 - 3.14e12T^{2} \)
67 \( 1 - 6.06e12T^{2} \)
71 \( 1 + 9.09e12T^{2} \)
73 \( 1 + 2.81e6T + 1.10e13T^{2} \)
79 \( 1 + 7.56e6T + 1.92e13T^{2} \)
83 \( 1 + 8.36e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 + 1.17e7T + 8.07e13T^{2} \)
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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

−12.46134326539530540001033827274, −11.45932674267913354402041858448, −9.998866218785554060420595236770, −8.925775764603181529601169411735, −8.341158301292692795896627008245, −5.81567048771579104032684665277, −4.66960483733547722619271184263, −3.27455762753789063951133729206, −1.25733569173564249618206380507, −0.27817727734191117039697205862, 2.68888608667737095455828658446, 4.16012687790846032521296712357, 6.14707998115043803754847396476, 6.86620070201066839629689654187, 7.77588969726576338970570623994, 9.742452573279157819528930403602, 10.20116296391575939678359287549, 11.87209134335341888149725856594, 13.30011263114948818579681356481, 14.24863772841221404555557794180

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