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Properties

Label 2-72-8.3-c2-0-1
Degree $2$
Conductor $72$
Sign $0.774 - 0.632i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 1.46i)2-s + (−0.267 + 3.99i)4-s + 7.98i·5-s + 2.13i·7-s + (6.19 − 5.06i)8-s + (11.6 − 10.9i)10-s + 8·11-s + 11.6i·13-s + (3.12 − 2.92i)14-s + (−15.8 − 2.13i)16-s − 11.8·17-s + 14.9·19-s + (−31.8 − 2.13i)20-s + (−10.9 − 11.6i)22-s + 4.27i·23-s + ⋯
L(s)  = 1  + (−0.683 − 0.730i)2-s + (−0.0669 + 0.997i)4-s + 1.59i·5-s + 0.305i·7-s + (0.774 − 0.632i)8-s + (1.16 − 1.09i)10-s + 0.727·11-s + 0.898i·13-s + (0.223 − 0.208i)14-s + (−0.991 − 0.133i)16-s − 0.697·17-s + 0.785·19-s + (−1.59 − 0.106i)20-s + (−0.496 − 0.531i)22-s + 0.185i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.774 - 0.632i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.809769 + 0.288653i\)
\(L(\frac12)\) \(\approx\) \(0.809769 + 0.288653i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 1.46i)T \)
3 \( 1 \)
good5 \( 1 - 7.98iT - 25T^{2} \)
7 \( 1 - 2.13iT - 49T^{2} \)
11 \( 1 - 8T + 121T^{2} \)
13 \( 1 - 11.6iT - 169T^{2} \)
17 \( 1 + 11.8T + 289T^{2} \)
19 \( 1 - 14.9T + 361T^{2} \)
23 \( 1 - 4.27iT - 529T^{2} \)
29 \( 1 + 0.573iT - 841T^{2} \)
31 \( 1 + 57.4iT - 961T^{2} \)
37 \( 1 - 27.6iT - 1.36e3T^{2} \)
41 \( 1 - 31.5T + 1.68e3T^{2} \)
43 \( 1 - 28.7T + 1.84e3T^{2} \)
47 \( 1 + 59.5iT - 2.20e3T^{2} \)
53 \( 1 + 31.3iT - 2.80e3T^{2} \)
59 \( 1 - 52.7T + 3.48e3T^{2} \)
61 \( 1 + 59.5iT - 3.72e3T^{2} \)
67 \( 1 + 84.7T + 4.48e3T^{2} \)
71 \( 1 - 42.4iT - 5.04e3T^{2} \)
73 \( 1 + 5.42T + 5.32e3T^{2} \)
79 \( 1 - 44.6iT - 6.24e3T^{2} \)
83 \( 1 + 67.7T + 6.88e3T^{2} \)
89 \( 1 - 133.T + 7.92e3T^{2} \)
97 \( 1 - 97.1T + 9.40e3T^{2} \)
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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

−14.43244951597317680512210083523, −13.41475164075039643012502703397, −11.76139397506278485834806280415, −11.27984212606822627611752612148, −10.06810863887070178928168700290, −9.080282463822533586893818136635, −7.49800902505051585275999164462, −6.48455552306572344912327671009, −3.86097316195256239732338945321, −2.36620447646403876290220758775, 1.03992153995220372661567779604, 4.57227852994403919594246230971, 5.77324905175859997564278806102, 7.39061357997135212246354669911, 8.622317305641944247219840581777, 9.318578485766785567438175810599, 10.66160014150586465012153707820, 12.16612926128509256214956913106, 13.29531537388394532190326164353, 14.38363123878087061069384512078

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