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Properties

Label 2-72-72.13-c1-0-6
Degree $2$
Conductor $72$
Sign $0.819 + 0.573i$
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  + (0.366 − 1.36i)2-s + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (1.73 + i)5-s + (2.36 − 0.633i)6-s + (−2 − 3.46i)7-s + (−2 + 1.99i)8-s + (−1.5 + 2.59i)9-s + (2 − 1.99i)10-s + (−2.59 + 1.5i)11-s − 3.46i·12-s + (−1.73 − i)13-s + (−5.46 + 1.46i)14-s + 3.46i·15-s + (1.99 + 3.46i)16-s + 5·17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (0.774 + 0.447i)5-s + (0.965 − 0.258i)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 + (−1.46 + 0.391i)14-s + 0.894i·15-s + (0.499 + 0.866i)16-s + 1.21·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.819 + 0.573i$
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: \((2,\ 72,\ (\ :1/2),\ 0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04333 - 0.328961i\)
\(L(\frac12)\) \(\approx\) \(1.04333 - 0.328961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
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

−14.13727068886058197892349261299, −13.67445576818653449017017779628, −12.49059798390126917928632934601, −10.72570299456177282736360006327, −10.13251765185218936814101371061, −9.557197029839213090619546303397, −7.69991990660725264961026418096, −5.62945459857429237262547075789, −4.08380052048807056291574053817, −2.75148535640928773077850167267, 2.87052640885778066793988540552, 5.44469430329803475229840398074, 6.24321063915552237522606768556, 7.72766623118291055585766666734, 8.864233791200871868373160028945, 9.640827493928599083202869088148, 12.18207237333949929815455512245, 12.77753997839591242264394960293, 13.70206001225455086377290891423, 14.65225774412273451644233915024

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