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Properties

Label 2-60-60.59-c3-0-24
Degree $2$
Conductor $60$
Sign $-0.364 + 0.931i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.40 + 1.49i)2-s + (3.96 − 3.36i)3-s + (3.54 − 7.17i)4-s + (−9.10 − 6.48i)5-s + (−4.49 + 13.9i)6-s − 26.5·7-s + (2.20 + 22.5i)8-s + (4.37 − 26.6i)9-s + (31.5 + 1.96i)10-s − 17.2·11-s + (−10.1 − 40.3i)12-s − 66.7i·13-s + (63.7 − 39.6i)14-s + (−57.8 + 4.95i)15-s + (−38.9 − 50.7i)16-s + 87.8·17-s + ⋯
L(s)  = 1  + (−0.849 + 0.527i)2-s + (0.762 − 0.647i)3-s + (0.442 − 0.896i)4-s + (−0.814 − 0.579i)5-s + (−0.305 + 0.952i)6-s − 1.43·7-s + (0.0976 + 0.995i)8-s + (0.162 − 0.986i)9-s + (0.998 + 0.0622i)10-s − 0.471·11-s + (−0.243 − 0.970i)12-s − 1.42i·13-s + (1.21 − 0.756i)14-s + (−0.996 + 0.0852i)15-s + (−0.608 − 0.793i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.364 + 0.931i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.364 + 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.387519 - 0.567776i\)
\(L(\frac12)\) \(\approx\) \(0.387519 - 0.567776i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.40 - 1.49i)T \)
3 \( 1 + (-3.96 + 3.36i)T \)
5 \( 1 + (9.10 + 6.48i)T \)
good7 \( 1 + 26.5T + 343T^{2} \)
11 \( 1 + 17.2T + 1.33e3T^{2} \)
13 \( 1 + 66.7iT - 2.19e3T^{2} \)
17 \( 1 - 87.8T + 4.91e3T^{2} \)
19 \( 1 - 22.3iT - 6.85e3T^{2} \)
23 \( 1 + 30.5iT - 1.21e4T^{2} \)
29 \( 1 + 54.3iT - 2.43e4T^{2} \)
31 \( 1 - 143. iT - 2.97e4T^{2} \)
37 \( 1 + 285. iT - 5.06e4T^{2} \)
41 \( 1 + 57.8iT - 6.89e4T^{2} \)
43 \( 1 - 284.T + 7.95e4T^{2} \)
47 \( 1 - 111. iT - 1.03e5T^{2} \)
53 \( 1 + 160.T + 1.48e5T^{2} \)
59 \( 1 + 447.T + 2.05e5T^{2} \)
61 \( 1 - 559.T + 2.26e5T^{2} \)
67 \( 1 - 84.2T + 3.00e5T^{2} \)
71 \( 1 - 119.T + 3.57e5T^{2} \)
73 \( 1 - 253. iT - 3.89e5T^{2} \)
79 \( 1 + 1.18e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + 859. iT - 7.04e5T^{2} \)
97 \( 1 + 79.1iT - 9.12e5T^{2} \)
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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

−14.48595261000734475011259004661, −13.00334576349979155650692116457, −12.26997754419583078534253059416, −10.37849858199738675051609352960, −9.280500156897373485287563649383, −8.119858813063408223039336971150, −7.33175210757199662993571157568, −5.81328107434897422873504581021, −3.17220321574512626772639590320, −0.57044982137813009137022668126, 2.83991967094821038034172599485, 3.89555473595841977771100945292, 6.84011304160879656711281561016, 7.997087922396596476518328256136, 9.340581751785863835096437012539, 10.07401304528098772066196659838, 11.26593854087772835254929342600, 12.49607729935204415405505921034, 13.80492039520802256717258355999, 15.23225975296404099148602743609

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