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Properties

Label 2-72-72.11-c3-0-27
Degree $2$
Conductor $72$
Sign $-0.484 + 0.874i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.31 − 1.62i)2-s + (−3.80 + 3.53i)3-s + (2.71 − 7.52i)4-s + (−10.8 − 18.8i)5-s + (−3.06 + 14.3i)6-s + (10.5 + 6.09i)7-s + (−5.94 − 21.8i)8-s + (1.99 − 26.9i)9-s + (−55.7 − 25.8i)10-s + (−22.8 − 13.1i)11-s + (16.2 + 38.2i)12-s + (24.0 − 13.8i)13-s + (34.3 − 3.05i)14-s + (107. + 33.2i)15-s + (−49.2 − 40.8i)16-s + 56.3i·17-s + ⋯
L(s)  = 1  + (0.818 − 0.574i)2-s + (−0.732 + 0.680i)3-s + (0.339 − 0.940i)4-s + (−0.971 − 1.68i)5-s + (−0.208 + 0.977i)6-s + (0.570 + 0.329i)7-s + (−0.262 − 0.964i)8-s + (0.0739 − 0.997i)9-s + (−1.76 − 0.819i)10-s + (−0.625 − 0.360i)11-s + (0.391 + 0.920i)12-s + (0.512 − 0.295i)13-s + (0.655 − 0.0582i)14-s + (1.85 + 0.572i)15-s + (−0.769 − 0.638i)16-s + 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.484 + 0.874i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.734735 - 1.24753i\)
\(L(\frac12)\) \(\approx\) \(0.734735 - 1.24753i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.31 + 1.62i)T \)
3 \( 1 + (3.80 - 3.53i)T \)
good5 \( 1 + (10.8 + 18.8i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-10.5 - 6.09i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (22.8 + 13.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.0 + 13.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 56.3iT - 4.91e3T^{2} \)
19 \( 1 - 80.9T + 6.85e3T^{2} \)
23 \( 1 + (-4.32 - 7.49i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-43.8 + 75.8i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-174. + 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 53.3iT - 5.06e4T^{2} \)
41 \( 1 + (-398. + 229. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (105. - 181. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-168. + 292. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 84.2T + 1.48e5T^{2} \)
59 \( 1 + (86.9 - 50.2i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (342. + 197. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-68.5 - 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 766.T + 3.57e5T^{2} \)
73 \( 1 - 464.T + 3.89e5T^{2} \)
79 \( 1 + (-906. - 523. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-1.13e3 - 654. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 403. iT - 7.04e5T^{2} \)
97 \( 1 + (29.9 - 51.9i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.45738685451499429765608125333, −12.41399960028852449070794135512, −11.74765631545565727939258174469, −10.83424466824820712595841535374, −9.386291695252423464246470518013, −8.106096927895219994190517536938, −5.74545007841943911182892263006, −4.90518790303534135936857277867, −3.84376773705442147173109237388, −0.842152391028523753694857443841, 2.90708981404567799799520148492, 4.65698878347123824255709660036, 6.29868419958830746409961277059, 7.30604688044857598003691199457, 7.85458313031624519090617835536, 10.66067384377548549326559022109, 11.40221655955405469087426617619, 12.18048333725423040135006405716, 13.68983624878213294211424338229, 14.34493088990301493287836249141

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