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Properties

Label 2-72-72.61-c1-0-0
Degree $2$
Conductor $72$
Sign $-0.573 - 0.819i$
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  + (−1.36 + 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (−1.73 + i)5-s + (0.633 − 2.36i)6-s + (−2 + 3.46i)7-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (1.99 − 2i)10-s + (2.59 + 1.5i)11-s + 3.46i·12-s + (1.73 − i)13-s + (1.46 − 5.46i)14-s − 3.46i·15-s + (1.99 − 3.46i)16-s + 5·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (−0.774 + 0.447i)5-s + (0.258 − 0.965i)6-s + (−0.755 + 1.30i)7-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.632 − 0.632i)10-s + (0.783 + 0.452i)11-s + 0.999i·12-s + (0.480 − 0.277i)13-s + (0.391 − 1.46i)14-s − 0.894i·15-s + (0.499 − 0.866i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ -0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199862 + 0.383933i\)
\(L(\frac12)\) \(\approx\) \(0.199862 + 0.383933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
good5 \( 1 + (1.73 - i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 - 6i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.29265418580066753997779227219, −14.67386581721776456450694028739, −12.17675600904718451274949134417, −11.67026146300814555242580423990, −10.34326000840739451213141599018, −9.443369303386711787844930720142, −8.370991586526766644620930897732, −6.70322670223246431719646270721, −5.59999673995470352727593848927, −3.32942124861589429372848992180, 0.884195755745734177353933001764, 3.70685003664035112911932368740, 6.30334462187447895786850271729, 7.35564051714940854583092676864, 8.289693896764442110070953991204, 9.804990840672986192092784488114, 11.06091080108340116739538204240, 11.89287259888521964653672163956, 12.87709214913254813933916515271, 14.03390117928283652825979833217

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