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Properties

Label 2-72-72.59-c1-0-9
Degree $2$
Conductor $72$
Sign $-0.329 + 0.944i$
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  + (0.409 − 1.35i)2-s + (−1.12 − 1.31i)3-s + (−1.66 − 1.10i)4-s + (−0.565 + 0.978i)5-s + (−2.24 + 0.988i)6-s + (3.71 − 2.14i)7-s + (−2.18 + 1.79i)8-s + (−0.456 + 2.96i)9-s + (1.09 + 1.16i)10-s + (1.00 − 0.582i)11-s + (0.419 + 3.43i)12-s + (2.64 + 1.52i)13-s + (−1.38 − 5.90i)14-s + (1.92 − 0.360i)15-s + (1.54 + 3.69i)16-s − 1.49i·17-s + ⋯
L(s)  = 1  + (0.289 − 0.957i)2-s + (−0.651 − 0.758i)3-s + (−0.832 − 0.554i)4-s + (−0.252 + 0.437i)5-s + (−0.915 + 0.403i)6-s + (1.40 − 0.810i)7-s + (−0.771 + 0.636i)8-s + (−0.152 + 0.988i)9-s + (0.345 + 0.368i)10-s + (0.304 − 0.175i)11-s + (0.121 + 0.992i)12-s + (0.733 + 0.423i)13-s + (−0.369 − 1.57i)14-s + (0.496 − 0.0932i)15-s + (0.385 + 0.922i)16-s − 0.362i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.329 + 0.944i$
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: \((2,\ 72,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503793 - 0.709479i\)
\(L(\frac12)\) \(\approx\) \(0.503793 - 0.709479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.409 + 1.35i)T \)
3 \( 1 + (1.12 + 1.31i)T \)
good5 \( 1 + (0.565 - 0.978i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.71 + 2.14i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.00 + 0.582i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.64 - 1.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 + 3.42T + 19T^{2} \)
23 \( 1 + (3.85 - 6.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.709 + 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.66 + 2.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.97iT - 37T^{2} \)
41 \( 1 + (4.23 + 2.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.74 + 3.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 + 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (-7.50 - 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.58 - 9.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + (2.24 - 1.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.98 - 2.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.63iT - 89T^{2} \)
97 \( 1 + (-3.35 - 5.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.93035407755272585140931107040, −13.28362393395700364297206153459, −11.69542468693414966146258707567, −11.34870139642146937120036835624, −10.38996401748245505406927314155, −8.515452925765671979536433076908, −7.21319963037127785582048995754, −5.54589142181019492760130056483, −4.06632531188568629328634915562, −1.62792039028054742034256897381, 4.18520632964577858566156115719, 5.17062500644679992682256880194, 6.31876240618624453363936905294, 8.238223075517799309705814316295, 8.875515933288566899556320357171, 10.58436726241657733162589750484, 11.85539473569968757729645079643, 12.69496720952127909143360765253, 14.45517755311644382171344605744, 14.97978302154472165615105386294

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