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Properties

Label 2-72-72.59-c1-0-0
Degree $2$
Conductor $72$
Sign $-0.932 - 0.360i$
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  + (−0.608 + 1.27i)2-s + (−1.71 + 0.231i)3-s + (−1.25 − 1.55i)4-s + (−1.74 + 3.01i)5-s + (0.748 − 2.33i)6-s + (−1.80 + 1.04i)7-s + (2.75 − 0.660i)8-s + (2.89 − 0.795i)9-s + (−2.79 − 4.06i)10-s + (−0.116 + 0.0675i)11-s + (2.52 + 2.37i)12-s + (2.63 + 1.52i)13-s + (−0.231 − 2.94i)14-s + (2.29 − 5.58i)15-s + (−0.830 + 3.91i)16-s + 4.19i·17-s + ⋯
L(s)  = 1  + (−0.430 + 0.902i)2-s + (−0.990 + 0.133i)3-s + (−0.629 − 0.777i)4-s + (−0.779 + 1.35i)5-s + (0.305 − 0.952i)6-s + (−0.683 + 0.394i)7-s + (0.972 − 0.233i)8-s + (0.964 − 0.265i)9-s + (−0.883 − 1.28i)10-s + (−0.0352 + 0.0203i)11-s + (0.727 + 0.685i)12-s + (0.731 + 0.422i)13-s + (−0.0619 − 0.786i)14-s + (0.591 − 1.44i)15-s + (−0.207 + 0.978i)16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0734457 + 0.394335i\)
\(L(\frac12)\) \(\approx\) \(0.0734457 + 0.394335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 - 1.27i)T \)
3 \( 1 + (1.71 - 0.231i)T \)
good5 \( 1 + (1.74 - 3.01i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.80 - 1.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.116 - 0.0675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.63 - 1.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 - 0.919T + 19T^{2} \)
23 \( 1 + (0.689 - 1.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.24 + 7.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.39 - 2.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 + (-1.79 - 1.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.41 - 9.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.205 - 0.356i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.968T + 53T^{2} \)
59 \( 1 + (-3.88 - 2.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 + 4.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.15 + 5.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (10.8 - 6.27i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.23 + 3.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 + 0.826i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.51696265619165483476802732301, −14.45189246662506312492286272234, −13.01272984312161346567660095415, −11.52697555067362037596343642602, −10.65620622063244349373937613232, −9.596736430884011093391556177854, −7.88534332612652737367525116376, −6.66940874963970156513598629788, −5.97572994778091420032943754136, −3.98966305529048461110879331557, 0.73549784888568791311413908671, 3.90969286045348549593597225416, 5.20723990760421266340880355064, 7.27115535437815740836172397684, 8.587903196434985901034352347965, 9.792387384558219904766919260241, 11.01261926034738595552759382896, 11.96672221569593507097049883167, 12.74184592215779855643027629883, 13.49394757874877752798320310684

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