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Properties

Label 2-72-72.13-c1-0-4
Degree $2$
Conductor $72$
Sign $0.923 - 0.383i$
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  + (1.41 − 0.0174i)2-s + (−0.294 + 1.70i)3-s + (1.99 − 0.0493i)4-s + (−3.17 − 1.83i)5-s + (−0.386 + 2.41i)6-s + (−0.191 − 0.332i)7-s + (2.82 − 0.104i)8-s + (−2.82 − 1.00i)9-s + (−4.51 − 2.53i)10-s + (1.73 − 1.00i)11-s + (−0.504 + 3.42i)12-s + (0.397 + 0.229i)13-s + (−0.277 − 0.466i)14-s + (4.06 − 4.87i)15-s + (3.99 − 0.197i)16-s − 4.08·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0123i)2-s + (−0.170 + 0.985i)3-s + (0.999 − 0.0246i)4-s + (−1.41 − 0.819i)5-s + (−0.157 + 0.987i)6-s + (−0.0725 − 0.125i)7-s + (0.999 − 0.0370i)8-s + (−0.942 − 0.335i)9-s + (−1.42 − 0.801i)10-s + (0.524 − 0.302i)11-s + (−0.145 + 0.989i)12-s + (0.110 + 0.0636i)13-s + (−0.0740 − 0.124i)14-s + (1.04 − 1.25i)15-s + (0.998 − 0.0493i)16-s − 0.990·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.923 - 0.383i$
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: \((2,\ 72,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24541 + 0.248190i\)
\(L(\frac12)\) \(\approx\) \(1.24541 + 0.248190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0174i)T \)
3 \( 1 + (0.294 - 1.70i)T \)
good5 \( 1 + (3.17 + 1.83i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.191 + 0.332i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 1.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.397 - 0.229i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + (2.97 - 5.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.03 + 1.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.592 + 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.74iT - 37T^{2} \)
41 \( 1 + (-4.75 + 8.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.03 - 0.598i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.27 - 5.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.63iT - 53T^{2} \)
59 \( 1 + (0.603 + 0.348i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.23 - 2.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.87 + 5.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (5.35 + 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.49 + 3.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.56T + 89T^{2} \)
97 \( 1 + (2.98 + 5.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.89158768020518181280371573564, −13.73795029026628181062716497944, −12.28079176520206254000149399241, −11.65073905923323373441152920978, −10.65160500322245493798257549729, −8.980611318041427731358515288423, −7.66860779871621928167793125357, −5.86771766523315102332136169218, −4.40993694174807581063971495976, −3.70635646021338764309467466835, 2.79090460979376617458335126158, 4.43438005215861515822575523786, 6.44660074858413157343182483825, 7.12185882965803515747559078268, 8.318436305652812604653358335439, 10.80955131623586879761898852606, 11.58836029851524616553771239781, 12.32504869227748441413575245328, 13.44861637841913998254751602052, 14.56413408495261936650150679258

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