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Properties

Label 2-72-24.5-c2-0-6
Degree $2$
Conductor $72$
Sign $0.321 + 0.946i$
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  + (1.16 − 1.62i)2-s + (−1.30 − 3.78i)4-s + 7.67·5-s − 7.21·7-s + (−7.67 − 2.27i)8-s + (8.90 − 12.4i)10-s + 6.05·11-s − 2.29i·13-s + (−8.37 + 11.7i)14-s + (−12.6 + 9.85i)16-s + 21.8i·17-s + 34.8i·19-s + (−9.99 − 29.0i)20-s + (7.02 − 9.85i)22-s − 21.5i·23-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.325 − 0.945i)4-s + 1.53·5-s − 1.03·7-s + (−0.958 − 0.283i)8-s + (0.890 − 1.24i)10-s + 0.550·11-s − 0.176i·13-s + (−0.598 + 0.838i)14-s + (−0.787 + 0.615i)16-s + 1.28i·17-s + 1.83i·19-s + (−0.499 − 1.45i)20-s + (0.319 − 0.447i)22-s − 0.935i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.321 + 0.946i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.321 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41216 - 1.01145i\)
\(L(\frac12)\) \(\approx\) \(1.41216 - 1.01145i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 1.62i)T \)
3 \( 1 \)
good5 \( 1 - 7.67T + 25T^{2} \)
7 \( 1 + 7.21T + 49T^{2} \)
11 \( 1 - 6.05T + 121T^{2} \)
13 \( 1 + 2.29iT - 169T^{2} \)
17 \( 1 - 21.8iT - 289T^{2} \)
19 \( 1 - 34.8iT - 361T^{2} \)
23 \( 1 + 21.5iT - 529T^{2} \)
29 \( 1 - 10.9T + 841T^{2} \)
31 \( 1 + 37.6T + 961T^{2} \)
37 \( 1 + 34.8iT - 1.36e3T^{2} \)
41 \( 1 + 13.3iT - 1.68e3T^{2} \)
43 \( 1 + 60.5iT - 1.84e3T^{2} \)
47 \( 1 - 3.34iT - 2.20e3T^{2} \)
53 \( 1 + 35.1T + 2.80e3T^{2} \)
59 \( 1 - 37.1T + 3.48e3T^{2} \)
61 \( 1 + 25.6iT - 3.72e3T^{2} \)
67 \( 1 - 25.6iT - 4.48e3T^{2} \)
71 \( 1 + 37.2iT - 5.04e3T^{2} \)
73 \( 1 + 77.6T + 5.32e3T^{2} \)
79 \( 1 - 31.3T + 6.24e3T^{2} \)
83 \( 1 - 55.3T + 6.88e3T^{2} \)
89 \( 1 - 5.43iT - 7.92e3T^{2} \)
97 \( 1 + 52.8T + 9.40e3T^{2} \)
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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

−14.04421438068504251516977628329, −12.92476644644837378997713617806, −12.38947364928796407475605538381, −10.56785212961177947193982561317, −9.973014232730866770284764003396, −8.931587805085988872316595481756, −6.38124805760637859428391653241, −5.68776511745155046980242734231, −3.68495211804995054709706564341, −1.91142564045386729776039079357, 2.92611784155963784839975619419, 4.99274355783893844568540632475, 6.23308521978685461767690093848, 7.06431761619142136386235753823, 9.127397213493304218039963257061, 9.623336055121070420346471614478, 11.51916771013797011761983605958, 13.02452924705454074256708193476, 13.49372792681367089586677141654, 14.42070220873387593358067174948

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