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Properties

Label 2-72-8.3-c6-0-13
Degree $2$
Conductor $72$
Sign $0.994 + 0.106i$
Analytic cond. $16.5638$
Root an. cond. $4.06987$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.24 − 6.78i)2-s + (−27.9 − 57.5i)4-s + 206. i·5-s − 210. i·7-s + (−509. − 54.6i)8-s + (1.39e3 + 874. i)10-s + 2.26e3·11-s + 3.53e3i·13-s + (−1.42e3 − 893. i)14-s + (−2.53e3 + 3.22e3i)16-s + 3.48e3·17-s + 9.84e3·19-s + (1.18e4 − 5.76e3i)20-s + (9.60e3 − 1.53e4i)22-s − 2.09e3i·23-s + ⋯
L(s)  = 1  + (0.530 − 0.847i)2-s + (−0.437 − 0.899i)4-s + 1.64i·5-s − 0.613i·7-s + (−0.994 − 0.106i)8-s + (1.39 + 0.874i)10-s + 1.69·11-s + 1.61i·13-s + (−0.519 − 0.325i)14-s + (−0.618 + 0.786i)16-s + 0.709·17-s + 1.43·19-s + (1.48 − 0.720i)20-s + (0.901 − 1.44i)22-s − 0.172i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ 0.994 + 0.106i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.42361 - 0.129769i\)
\(L(\frac12)\) \(\approx\) \(2.42361 - 0.129769i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.24 + 6.78i)T \)
3 \( 1 \)
good5 \( 1 - 206. iT - 1.56e4T^{2} \)
7 \( 1 + 210. iT - 1.17e5T^{2} \)
11 \( 1 - 2.26e3T + 1.77e6T^{2} \)
13 \( 1 - 3.53e3iT - 4.82e6T^{2} \)
17 \( 1 - 3.48e3T + 2.41e7T^{2} \)
19 \( 1 - 9.84e3T + 4.70e7T^{2} \)
23 \( 1 + 2.09e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.01e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.37e3iT - 8.87e8T^{2} \)
37 \( 1 - 5.50e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.84e4T + 4.75e9T^{2} \)
43 \( 1 + 1.06e5T + 6.32e9T^{2} \)
47 \( 1 - 8.40e4iT - 1.07e10T^{2} \)
53 \( 1 + 8.03e4iT - 2.21e10T^{2} \)
59 \( 1 + 8.45e4T + 4.21e10T^{2} \)
61 \( 1 - 1.60e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.29e5T + 9.04e10T^{2} \)
71 \( 1 - 1.84e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.56e5T + 1.51e11T^{2} \)
79 \( 1 + 4.43e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.56e5T + 3.26e11T^{2} \)
89 \( 1 - 5.93e5T + 4.96e11T^{2} \)
97 \( 1 + 7.83e5T + 8.32e11T^{2} \)
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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

−13.79762394061170472200735439487, −11.81303749976370705367281905097, −11.46959737775888814066081559628, −10.19499961692418016999668887199, −9.335949286028132311843658699639, −7.10902638565815706130692701403, −6.25432239428865567018172500048, −4.18624379099169120539420938287, −3.16576300441272644984560497789, −1.46622414986951500560734015138, 0.938484869980688737723732990709, 3.57233963432040769620742952589, 5.06787873481715337636049308823, 5.85008037524801322730950394390, 7.63163358784100234919573865691, 8.725278613234709637435081575507, 9.491444635413232828332419636454, 11.89916421429986986190241945170, 12.39989813734975513010002484879, 13.42366208178851715582317676237

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