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Properties

Label 2-72-9.4-c1-0-0
Degree $2$
Conductor $72$
Sign $0.766 - 0.642i$
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  + 1.73i·3-s + (0.5 − 0.866i)5-s + (1.5 + 2.59i)7-s − 2.99·9-s + (−2.5 − 4.33i)11-s + (2.5 − 4.33i)13-s + (1.49 + 0.866i)15-s − 2·17-s − 4·19-s + (−4.5 + 2.59i)21-s + (0.5 − 0.866i)23-s + (2 + 3.46i)25-s − 5.19i·27-s + (4.5 + 7.79i)29-s + (0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.223 − 0.387i)5-s + (0.566 + 0.981i)7-s − 0.999·9-s + (−0.753 − 1.30i)11-s + (0.693 − 1.20i)13-s + (0.387 + 0.223i)15-s − 0.485·17-s − 0.917·19-s + (−0.981 + 0.566i)21-s + (0.104 − 0.180i)23-s + (0.400 + 0.692i)25-s − 0.999i·27-s + (0.835 + 1.44i)29-s + (0.0898 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886588 + 0.322691i\)
\(L(\frac12)\) \(\approx\) \(0.886588 + 0.322691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.04596314949755698763470883040, −13.77072857964450116476589777167, −12.60509325931048725117885152786, −11.15093858439666734076360950424, −10.49873156591307052267971904906, −8.793596843996729558892910973048, −8.420518511795351127598420248194, −5.88309429526529991076949664900, −5.03124924463530726898553060118, −3.07728264120660582400431114899, 2.06254548776481924300575113000, 4.48286581092201091310744652728, 6.46187784115568010412385152438, 7.32039019467316323887690579853, 8.534045389151660698350575117265, 10.22739584430604561091129408885, 11.25386671404890912983066860805, 12.44378222695289156138008676434, 13.58160528459214923618533362032, 14.19464943602577262242666871189

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