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Properties

Label 2-72-72.13-c1-0-2
Degree $2$
Conductor $72$
Sign $0.791 - 0.610i$
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  + (−1.34 + 0.436i)2-s + (1.52 + 0.816i)3-s + (1.61 − 1.17i)4-s + (−0.602 − 0.348i)5-s + (−2.41 − 0.431i)6-s + (0.795 + 1.37i)7-s + (−1.66 + 2.28i)8-s + (1.66 + 2.49i)9-s + (0.962 + 0.205i)10-s + (2.37 − 1.36i)11-s + (3.43 − 0.470i)12-s + (−4.76 − 2.75i)13-s + (−1.67 − 1.50i)14-s + (−0.636 − 1.02i)15-s + (1.24 − 3.80i)16-s − 5.65·17-s + ⋯
L(s)  = 1  + (−0.951 + 0.308i)2-s + (0.882 + 0.471i)3-s + (0.809 − 0.586i)4-s + (−0.269 − 0.155i)5-s + (−0.984 − 0.176i)6-s + (0.300 + 0.520i)7-s + (−0.589 + 0.807i)8-s + (0.555 + 0.831i)9-s + (0.304 + 0.0649i)10-s + (0.715 − 0.412i)11-s + (0.990 − 0.135i)12-s + (−1.32 − 0.763i)13-s + (−0.446 − 0.402i)14-s + (−0.164 − 0.264i)15-s + (0.311 − 0.950i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.791 - 0.610i$
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: \((2,\ 72,\ (\ :1/2),\ 0.791 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742676 + 0.253136i\)
\(L(\frac12)\) \(\approx\) \(0.742676 + 0.253136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.436i)T \)
3 \( 1 + (-1.52 - 0.816i)T \)
good5 \( 1 + (0.602 + 0.348i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.795 - 1.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.37 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.76 + 2.75i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 0.963iT - 19T^{2} \)
23 \( 1 + (-3.28 + 5.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.85 + 1.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.69 - 6.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.25iT - 37T^{2} \)
41 \( 1 + (0.931 - 1.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.99 - 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.85 - 6.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.54iT - 53T^{2} \)
59 \( 1 + (4.62 + 2.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.93 + 4.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.95 - 3.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + (-2.87 - 4.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.74 - 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (1.24 + 2.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.01767830975755357782350099009, −14.13694678763154232469931892792, −12.46192828103001619322008529334, −11.13082072871743772722909188224, −9.992085852300930481645130156985, −8.880465908744216968842542154310, −8.193867575294066097225624339531, −6.79504590041498687806396072087, −4.85663134523370781838398484345, −2.57014684954820428635206915791, 2.04116080153799008088038296487, 3.93334731717163109013865403712, 6.95109993138513701164004883467, 7.48668813279105744966084957247, 8.948149587374605967982285121504, 9.692148190626114705223404215515, 11.21324480270749118581905264935, 12.17191399222121318615344336683, 13.39513652513261198998576543838, 14.66702673473994531238952445891

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