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Properties

Label 2-72-72.29-c2-0-11
Degree $2$
Conductor $72$
Sign $0.999 + 0.0398i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.50 − 1.31i)2-s + (0.102 + 2.99i)3-s + (0.519 − 3.96i)4-s + (3.47 + 6.02i)5-s + (4.10 + 4.37i)6-s + (2.29 − 3.97i)7-s + (−4.45 − 6.64i)8-s + (−8.97 + 0.614i)9-s + (13.1 + 4.46i)10-s + (7.77 − 13.4i)11-s + (11.9 + 1.15i)12-s + (−19.3 + 11.1i)13-s + (−1.79 − 9.00i)14-s + (−17.6 + 11.0i)15-s + (−15.4 − 4.12i)16-s − 9.17i·17-s + ⋯
L(s)  = 1  + (0.751 − 0.659i)2-s + (0.0341 + 0.999i)3-s + (0.129 − 0.991i)4-s + (0.695 + 1.20i)5-s + (0.684 + 0.728i)6-s + (0.327 − 0.567i)7-s + (−0.556 − 0.830i)8-s + (−0.997 + 0.0683i)9-s + (1.31 + 0.446i)10-s + (0.706 − 1.22i)11-s + (0.995 + 0.0959i)12-s + (−1.49 + 0.860i)13-s + (−0.128 − 0.643i)14-s + (−1.17 + 0.736i)15-s + (−0.966 − 0.257i)16-s − 0.539i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0398i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.999 + 0.0398i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.999 + 0.0398i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.87681 - 0.0373963i\)
\(L(\frac12)\) \(\approx\) \(1.87681 - 0.0373963i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.50 + 1.31i)T \)
3 \( 1 + (-0.102 - 2.99i)T \)
good5 \( 1 + (-3.47 - 6.02i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.29 + 3.97i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.77 + 13.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (19.3 - 11.1i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 9.17iT - 289T^{2} \)
19 \( 1 - 4.53iT - 361T^{2} \)
23 \( 1 + (2.69 - 1.55i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (6.42 - 11.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (2.17 + 3.77i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 23.0iT - 1.36e3T^{2} \)
41 \( 1 + (-60.6 + 35.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (51.7 + 29.8i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-32.1 - 18.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 24.3T + 2.80e3T^{2} \)
59 \( 1 + (-20.7 - 35.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.6 - 19.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24.3 + 14.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 59.8iT - 5.04e3T^{2} \)
73 \( 1 + 53.8T + 5.32e3T^{2} \)
79 \( 1 + (0.557 - 0.965i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (39.8 - 69.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 + (72.6 - 125. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.21658073227313871226500156925, −13.84750864704040597068203363847, −11.85072355490187559766394370182, −11.01429489151118449288310110602, −10.17277876814919051531078858510, −9.246921263130608045473529016701, −6.88460089232448237391337569742, −5.56934864583762673801049874580, −4.06818672885129355423950205062, −2.68336381590460289460019606390, 2.17511093762911943229039343114, 4.80611874917538689424458714264, 5.79603379699335531550484932129, 7.21489445100091957597689988523, 8.333389711259239224384908941858, 9.486543596929055034265100160021, 11.83736964538996897729378425792, 12.57638375235687220712824447455, 13.06790563226367637228539410926, 14.42996698662421643280544523244

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