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Properties

Label 2-72-72.59-c1-0-3
Degree $2$
Conductor $72$
Sign $0.870 - 0.491i$
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  + (−1.22 + 0.707i)2-s + (1.72 + 0.158i)3-s + (0.999 − 1.73i)4-s + (−2.22 + 1.02i)6-s + 2.82i·8-s + (2.94 + 0.548i)9-s + (−3.27 + 1.89i)11-s + (1.99 − 2.82i)12-s + (−2.00 − 3.46i)16-s − 8.02i·17-s + (−3.99 + 1.41i)18-s − 8.34·19-s + (2.67 − 4.63i)22-s + (−0.449 + 4.87i)24-s + (2.5 + 4.33i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.995 + 0.0917i)3-s + (0.499 − 0.866i)4-s + (−0.908 + 0.418i)6-s + 0.999i·8-s + (0.983 + 0.182i)9-s + (−0.987 + 0.570i)11-s + (0.577 − 0.816i)12-s + (−0.500 − 0.866i)16-s − 1.94i·17-s + (−0.942 + 0.333i)18-s − 1.91·19-s + (0.570 − 0.987i)22-s + (−0.0917 + 0.995i)24-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.870 - 0.491i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.870 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797740 + 0.209813i\)
\(L(\frac12)\) \(\approx\) \(0.797740 + 0.209813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (-1.72 - 0.158i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.27 - 1.89i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 8.02iT - 17T^{2} \)
19 \( 1 + 8.34T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (0.398 + 0.230i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-10.6 - 6.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.17 + 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.44 - 1.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (9.84 + 17.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.99026046468207165228230224067, −13.94325831746413774485719409405, −12.75793173445158789758196750730, −11.01267020733973071648887699286, −9.918566536362943500092950394430, −8.976847986167529653157356874982, −7.86177047040072258534126108663, −6.88674705314451405041949624189, −4.91148107712993364417179104961, −2.46497567617430314622651811016, 2.26572919897296598094005540027, 3.89150805359828389086771997983, 6.56921485253876360296646337170, 8.193271439978103121820558346488, 8.542133080835188017557302649565, 10.12341874679207186463613675070, 10.82257133950445822006569975190, 12.59712718113872897415425981215, 13.13253616871827371677520951235, 14.68592575309039060882362215601

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