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Properties

Label 2-72-72.13-c1-0-5
Degree $2$
Conductor $72$
Sign $0.657 - 0.753i$
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  + (0.587 + 1.28i)2-s + (1.69 − 0.378i)3-s + (−1.30 + 1.51i)4-s + (−1.97 − 1.14i)5-s + (1.47 + 1.95i)6-s + (−0.907 − 1.57i)7-s + (−2.71 − 0.795i)8-s + (2.71 − 1.27i)9-s + (0.306 − 3.21i)10-s + (−4.24 + 2.44i)11-s + (−1.64 + 3.05i)12-s + (4.00 + 2.31i)13-s + (1.48 − 2.09i)14-s + (−3.77 − 1.18i)15-s + (−0.570 − 3.95i)16-s + 1.92·17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.975 − 0.218i)3-s + (−0.654 + 0.755i)4-s + (−0.883 − 0.510i)5-s + (0.604 + 0.796i)6-s + (−0.343 − 0.594i)7-s + (−0.959 − 0.281i)8-s + (0.904 − 0.426i)9-s + (0.0968 − 1.01i)10-s + (−1.27 + 0.738i)11-s + (−0.473 + 0.880i)12-s + (1.11 + 0.641i)13-s + (0.398 − 0.559i)14-s + (−0.973 − 0.304i)15-s + (−0.142 − 0.989i)16-s + 0.467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.657 - 0.753i$
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: \((2,\ 72,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07287 + 0.487486i\)
\(L(\frac12)\) \(\approx\) \(1.07287 + 0.487486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 1.28i)T \)
3 \( 1 + (-1.69 + 0.378i)T \)
good5 \( 1 + (1.97 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.907 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.24 - 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.00 - 2.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.16 + 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.98iT - 37T^{2} \)
41 \( 1 + (2.36 - 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.02 + 3.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.95iT - 53T^{2} \)
59 \( 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 0.991i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.72 + 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (4.97 + 8.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.12 + 1.80i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.91802377515106952169637665344, −13.62865613062889630224978825829, −13.09422987969527482029447139767, −11.95254456046828480057570097692, −9.982080615838929807857780270381, −8.582089976265332008898049886723, −7.82605066229181761490246488452, −6.77717620251028244551546295518, −4.70070231611120625295778626878, −3.48748889216191682597875390263, 2.82368472877987340325548906309, 3.78299947695915591829500263405, 5.68312454052873255757110201877, 7.85702128582173897228789963487, 8.824375187601826531347937934966, 10.27357586118884874064143799284, 11.05199725887102069148058036778, 12.46950248130481375709006721605, 13.34373313837765158441049972729, 14.38617051101068374344458415225

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