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Properties

Label 2-72-72.13-c1-0-0
Degree $2$
Conductor $72$
Sign $0.780 - 0.625i$
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  + (−1.12 − 0.857i)2-s + (−0.986 + 1.42i)3-s + (0.530 + 1.92i)4-s + (1.19 + 0.687i)5-s + (2.32 − 0.756i)6-s + (1.80 + 3.12i)7-s + (1.05 − 2.62i)8-s + (−1.05 − 2.80i)9-s + (−0.750 − 1.79i)10-s + (−1.83 + 1.05i)11-s + (−3.26 − 1.14i)12-s + (0.887 + 0.512i)13-s + (0.648 − 5.06i)14-s + (−2.15 + 1.01i)15-s + (−3.43 + 2.04i)16-s + 0.808·17-s + ⋯
L(s)  = 1  + (−0.795 − 0.606i)2-s + (−0.569 + 0.822i)3-s + (0.265 + 0.964i)4-s + (0.532 + 0.307i)5-s + (0.951 − 0.308i)6-s + (0.682 + 1.18i)7-s + (0.373 − 0.927i)8-s + (−0.351 − 0.936i)9-s + (−0.237 − 0.567i)10-s + (−0.552 + 0.319i)11-s + (−0.943 − 0.330i)12-s + (0.246 + 0.142i)13-s + (0.173 − 1.35i)14-s + (−0.556 + 0.262i)15-s + (−0.859 + 0.511i)16-s + 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.780 - 0.625i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580153 + 0.203668i\)
\(L(\frac12)\) \(\approx\) \(0.580153 + 0.203668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.857i)T \)
3 \( 1 + (0.986 - 1.42i)T \)
good5 \( 1 + (-1.19 - 0.687i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.887 - 0.512i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.808T + 17T^{2} \)
19 \( 1 + 7.43iT - 19T^{2} \)
23 \( 1 + (1.65 - 2.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.71 + 4.45i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.26 + 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.01iT - 37T^{2} \)
41 \( 1 + (3.45 - 5.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.245 + 0.142i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.61 + 6.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.86iT - 53T^{2} \)
59 \( 1 + (7.06 + 4.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.31 - 3.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.43 - 1.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.69T + 71T^{2} \)
73 \( 1 - 0.409T + 73T^{2} \)
79 \( 1 + (0.0456 + 0.0790i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.40 + 1.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (2.76 + 4.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.21952706829976606096850248138, −13.53166600808430133469952833683, −12.01665530211521370327876638252, −11.39360455242264959614876868719, −10.23559757450405248928660590675, −9.362566061340320460999133950827, −8.226184304132018064253997206045, −6.33291618581270849892484453978, −4.76927107142401172170774118457, −2.61346717802520765215091249341, 1.39370703819009206755234613233, 5.10825847941792175693597171830, 6.30402937013770091687977154050, 7.58389875875770512154714067723, 8.391563639925794792829038577855, 10.23690474320414211821867262052, 10.84514798291571246132185927178, 12.32787657250998156177072740789, 13.77124707185705970988937924474, 14.26363920695015609383479336250

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