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Properties

Label 2-72-8.5-c17-0-39
Degree $2$
Conductor $72$
Sign $0.982 - 0.187i$
Analytic cond. $131.919$
Root an. cond. $11.4856$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (200. + 301. i)2-s + (−5.07e4 + 1.20e5i)4-s − 1.59e6i·5-s − 1.66e7·7-s + (−4.66e7 + 8.91e6i)8-s + (4.80e8 − 3.19e8i)10-s + 6.79e8i·11-s + 3.13e9i·13-s + (−3.32e9 − 5.00e9i)14-s + (−1.20e10 − 1.22e10i)16-s + 1.26e10·17-s − 5.87e9i·19-s + (1.92e11 + 8.08e10i)20-s + (−2.04e11 + 1.36e11i)22-s − 5.43e11·23-s + ⋯
L(s)  = 1  + (0.553 + 0.832i)2-s + (−0.387 + 0.921i)4-s − 1.82i·5-s − 1.08·7-s + (−0.982 + 0.187i)8-s + (1.51 − 1.00i)10-s + 0.955i·11-s + 1.06i·13-s + (−0.602 − 0.906i)14-s + (−0.700 − 0.714i)16-s + 0.440·17-s − 0.0794i·19-s + (1.68 + 0.705i)20-s + (−0.795 + 0.528i)22-s − 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.982 - 0.187i$
Analytic conductor: \(131.919\)
Root analytic conductor: \(11.4856\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :17/2),\ 0.982 - 0.187i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.623171565\)
\(L(\frac12)\) \(\approx\) \(1.623171565\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-200. - 301. i)T \)
3 \( 1 \)
good5 \( 1 + 1.59e6iT - 7.62e11T^{2} \)
7 \( 1 + 1.66e7T + 2.32e14T^{2} \)
11 \( 1 - 6.79e8iT - 5.05e17T^{2} \)
13 \( 1 - 3.13e9iT - 8.65e18T^{2} \)
17 \( 1 - 1.26e10T + 8.27e20T^{2} \)
19 \( 1 + 5.87e9iT - 5.48e21T^{2} \)
23 \( 1 + 5.43e11T + 1.41e23T^{2} \)
29 \( 1 - 2.82e12iT - 7.25e24T^{2} \)
31 \( 1 - 8.42e11T + 2.25e25T^{2} \)
37 \( 1 + 6.17e12iT - 4.56e26T^{2} \)
41 \( 1 - 6.38e13T + 2.61e27T^{2} \)
43 \( 1 + 5.41e13iT - 5.87e27T^{2} \)
47 \( 1 - 3.89e13T + 2.66e28T^{2} \)
53 \( 1 + 5.92e14iT - 2.05e29T^{2} \)
59 \( 1 + 7.11e14iT - 1.27e30T^{2} \)
61 \( 1 - 9.01e14iT - 2.24e30T^{2} \)
67 \( 1 + 8.96e14iT - 1.10e31T^{2} \)
71 \( 1 + 8.49e13T + 2.96e31T^{2} \)
73 \( 1 - 1.01e16T + 4.74e31T^{2} \)
79 \( 1 - 3.99e15T + 1.81e32T^{2} \)
83 \( 1 + 2.19e15iT - 4.21e32T^{2} \)
89 \( 1 - 6.74e16T + 1.37e33T^{2} \)
97 \( 1 + 1.67e15T + 5.95e33T^{2} \)
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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

−12.01490482541686686736749985681, −9.685706429592733915669428801814, −9.068185814977823779937681217486, −7.947076474895568669475900628670, −6.71978593647969946386728144413, −5.58393962121624823688670215131, −4.57883161885327159487583679081, −3.76066297196109764694991714717, −1.96031984063749782928881575817, −0.42741383060272953982741096859, 0.62092079999220958546154276527, 2.42322749921667785081192131928, 3.10933921773178314656718339560, 3.82803143398518535288241186014, 5.88110537726463015006280619908, 6.30890172888743781628095798311, 7.82771892399664348654752851623, 9.658288870871428167138765330239, 10.35013327963404641223506770913, 11.13174009945801384442928497050

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