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Properties

Label 2-72-24.11-c7-0-20
Degree $2$
Conductor $72$
Sign $-0.648 + 0.761i$
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  + (−11.3 − 0.338i)2-s + (127. + 7.64i)4-s + 93.0·5-s − 1.02e3i·7-s + (−1.44e3 − 129. i)8-s + (−1.05e3 − 31.4i)10-s + 1.54e3i·11-s − 5.67e3i·13-s + (−346. + 1.15e4i)14-s + (1.62e4 + 1.95e3i)16-s + 1.57e4i·17-s + 3.39e4·19-s + (1.18e4 + 712. i)20-s + (522. − 1.74e4i)22-s − 3.00e4·23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0298i)2-s + (0.998 + 0.0597i)4-s + 0.333·5-s − 1.12i·7-s + (−0.995 − 0.0895i)8-s + (−0.332 − 0.00995i)10-s + 0.350i·11-s − 0.716i·13-s + (−0.0337 + 1.12i)14-s + (0.992 + 0.119i)16-s + 0.775i·17-s + 1.13·19-s + (0.332 + 0.0199i)20-s + (0.0104 − 0.349i)22-s − 0.515·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.648 + 0.761i$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ -0.648 + 0.761i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.314889 - 0.681544i\)
\(L(\frac12)\) \(\approx\) \(0.314889 - 0.681544i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.3 + 0.338i)T \)
3 \( 1 \)
good5 \( 1 - 93.0T + 7.81e4T^{2} \)
7 \( 1 + 1.02e3iT - 8.23e5T^{2} \)
11 \( 1 - 1.54e3iT - 1.94e7T^{2} \)
13 \( 1 + 5.67e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.57e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.39e4T + 8.93e8T^{2} \)
23 \( 1 + 3.00e4T + 3.40e9T^{2} \)
29 \( 1 + 9.90e4T + 1.72e10T^{2} \)
31 \( 1 + 1.52e5iT - 2.75e10T^{2} \)
37 \( 1 + 3.36e5iT - 9.49e10T^{2} \)
41 \( 1 + 6.63e5iT - 1.94e11T^{2} \)
43 \( 1 + 7.07e5T + 2.71e11T^{2} \)
47 \( 1 + 4.02e5T + 5.06e11T^{2} \)
53 \( 1 + 6.62e5T + 1.17e12T^{2} \)
59 \( 1 + 1.74e6iT - 2.48e12T^{2} \)
61 \( 1 + 3.21e5iT - 3.14e12T^{2} \)
67 \( 1 + 4.64e5T + 6.06e12T^{2} \)
71 \( 1 - 3.53e6T + 9.09e12T^{2} \)
73 \( 1 + 3.41e6T + 1.10e13T^{2} \)
79 \( 1 - 7.60e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.11e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.00e7iT - 4.42e13T^{2} \)
97 \( 1 + 5.88e6T + 8.07e13T^{2} \)
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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

−12.68911708133927301800611591659, −11.35588469027084449233617595560, −10.31066082969443865759061611562, −9.580760567487230370475354128843, −8.044058108460701331284184388254, −7.18252210082952676401713310904, −5.75180024537511799392602258421, −3.64664712749910141787997482602, −1.80749559453904822608551703544, −0.35289382712463266125796662765, 1.57429548016659894165046624903, 2.94948630607963530940716606353, 5.40994648702204758601477642326, 6.58854127475798998870758667890, 8.005075467942219722932604100301, 9.146369958418283503618541659880, 9.876915723411373538604617811593, 11.44699741149896189889999035081, 12.01677491623647111819068572449, 13.64152412446762530503372996819

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