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Properties

Label 2-72-72.11-c1-0-5
Degree $2$
Conductor $72$
Sign $0.986 - 0.164i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.724 − 1.57i)3-s + (0.999 + 1.73i)4-s + (0.224 − 2.43i)6-s + 2.82i·8-s + (−1.94 + 2.28i)9-s + (−5.72 − 3.30i)11-s + (2 − 2.82i)12-s + (−2.00 + 3.46i)16-s + 2.36i·17-s + (−4 + 1.41i)18-s + 6.34·19-s + (−4.67 − 8.09i)22-s + (4.44 − 2.04i)24-s + (2.5 − 4.33i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.418 − 0.908i)3-s + (0.499 + 0.866i)4-s + (0.0917 − 0.995i)6-s + 0.999i·8-s + (−0.649 + 0.760i)9-s + (−1.72 − 0.996i)11-s + (0.577 − 0.816i)12-s + (−0.500 + 0.866i)16-s + 0.574i·17-s + (−0.942 + 0.333i)18-s + 1.45·19-s + (−0.996 − 1.72i)22-s + (0.908 − 0.418i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20457 + 0.0997875i\)
\(L(\frac12)\) \(\approx\) \(1.20457 + 0.0997875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (0.724 + 1.57i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.72 + 3.30i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.36iT - 17T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-9.39 + 5.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.17 - 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (1.62 - 0.937i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.174 + 0.301i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.44 - 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-4.84 + 8.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.40965945685779600935486563878, −13.46041437988458620714018389853, −12.80600452042692713469846878027, −11.66256123931092962844101076801, −10.65776290524712935405876278964, −8.323805465493111519116727905457, −7.51439919872580222149725574821, −6.10263812145192357411800228002, −5.14872745705443002593593979642, −2.87003400110696441359977158822, 3.00141070934747919454022704351, 4.73303866172930221653669861932, 5.54245330779234409972191726848, 7.33584879936360968511900110449, 9.487939876288600041548634650951, 10.32980201828229267574607016979, 11.31416558398520816406580231266, 12.38232301493602413733278662323, 13.46430198330321125217758542966, 14.70411958085262587315876972994

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