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Properties

Label 2-60-15.2-c3-0-2
Degree $2$
Conductor $60$
Sign $0.737 - 0.675i$
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  + (2.53 + 4.53i)3-s + (3.53 − 10.6i)5-s + (23 + 23i)7-s + (−14.1 + 23.0i)9-s + 7.07i·11-s + (−24 + 24i)13-s + (57.0 − 10.8i)15-s + (77.7 − 77.7i)17-s − 68i·19-s + (−46 + 162. i)21-s + (−98.9 − 98.9i)23-s + (−100. − 75i)25-s + (−140. − 5.82i)27-s − 134.·29-s + 94·31-s + ⋯
L(s)  = 1  + (0.487 + 0.872i)3-s + (0.316 − 0.948i)5-s + (1.24 + 1.24i)7-s + (−0.523 + 0.851i)9-s + 0.193i·11-s + (−0.512 + 0.512i)13-s + (0.982 − 0.186i)15-s + (1.10 − 1.10i)17-s − 0.821i·19-s + (−0.478 + 1.68i)21-s + (−0.897 − 0.897i)23-s + (−0.800 − 0.599i)25-s + (−0.999 − 0.0415i)27-s − 0.860·29-s + 0.544·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.66989 + 0.649209i\)
\(L(\frac12)\) \(\approx\) \(1.66989 + 0.649209i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.53 - 4.53i)T \)
5 \( 1 + (-3.53 + 10.6i)T \)
good7 \( 1 + (-23 - 23i)T + 343iT^{2} \)
11 \( 1 - 7.07iT - 1.33e3T^{2} \)
13 \( 1 + (24 - 24i)T - 2.19e3iT^{2} \)
17 \( 1 + (-77.7 + 77.7i)T - 4.91e3iT^{2} \)
19 \( 1 + 68iT - 6.85e3T^{2} \)
23 \( 1 + (98.9 + 98.9i)T + 1.21e4iT^{2} \)
29 \( 1 + 134.T + 2.43e4T^{2} \)
31 \( 1 - 94T + 2.97e4T^{2} \)
37 \( 1 + (66 + 66i)T + 5.06e4iT^{2} \)
41 \( 1 - 197. iT - 6.89e4T^{2} \)
43 \( 1 + (-126 + 126i)T - 7.95e4iT^{2} \)
47 \( 1 + (-84.8 + 84.8i)T - 1.03e5iT^{2} \)
53 \( 1 + (325. + 325. i)T + 1.48e5iT^{2} \)
59 \( 1 + 49.4T + 2.05e5T^{2} \)
61 \( 1 + 126T + 2.26e5T^{2} \)
67 \( 1 + (-68 - 68i)T + 3.00e5iT^{2} \)
71 \( 1 - 947. iT - 3.57e5T^{2} \)
73 \( 1 + (-403 + 403i)T - 3.89e5iT^{2} \)
79 \( 1 + 234iT - 4.93e5T^{2} \)
83 \( 1 + (-721. - 721. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + (-723 - 723i)T + 9.12e5iT^{2} \)
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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

−14.70624245509862972501893785310, −13.92505231819335263054133457260, −12.31260881155533867703165569189, −11.41865875275496383406385969395, −9.765799578263983459595110383434, −8.928066350205377395405411070621, −7.952205393944289730752800745774, −5.43065947221431415231453552066, −4.63801260232617228321022465402, −2.27544583151828959995699602713, 1.62406629612410404771903496360, 3.61262379811086959444258348236, 5.90624856868930834177048412532, 7.48051906809941823493057201800, 7.972899572610735843763209526625, 9.981223914231789123555085394601, 10.99234233191301903774868918853, 12.26287024890642841933920940564, 13.68764790975247352797299396085, 14.28692401934880655825662920480

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