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Properties

Label 2-60-15.14-c2-0-1
Degree $2$
Conductor $60$
Sign $0.929 - 0.368i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
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  + (2.23 + 2i)3-s + (2.23 − 4.47i)5-s + 8i·7-s + (1.00 + 8.94i)9-s − 8.94i·11-s − 12i·13-s + (13.9 − 5.52i)15-s − 31.3·17-s − 6·19-s + (−16 + 17.8i)21-s + 4.47·23-s + (−15.0 − 20.0i)25-s + (−15.6 + 22.0i)27-s − 26.8i·29-s + 34·31-s + ⋯
L(s)  = 1  + (0.745 + 0.666i)3-s + (0.447 − 0.894i)5-s + 1.14i·7-s + (0.111 + 0.993i)9-s − 0.813i·11-s − 0.923i·13-s + (0.929 − 0.368i)15-s − 1.84·17-s − 0.315·19-s + (−0.761 + 0.851i)21-s + 0.194·23-s + (−0.600 − 0.800i)25-s + (−0.579 + 0.814i)27-s − 0.925i·29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45178 + 0.277265i\)
\(L(\frac12)\) \(\approx\) \(1.45178 + 0.277265i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.23 - 2i)T \)
5 \( 1 + (-2.23 + 4.47i)T \)
good7 \( 1 - 8iT - 49T^{2} \)
11 \( 1 + 8.94iT - 121T^{2} \)
13 \( 1 + 12iT - 169T^{2} \)
17 \( 1 + 31.3T + 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 - 4.47T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 - 34T + 961T^{2} \)
37 \( 1 - 44iT - 1.36e3T^{2} \)
41 \( 1 - 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 28iT - 1.84e3T^{2} \)
47 \( 1 - 4.47T + 2.20e3T^{2} \)
53 \( 1 - 40.2T + 2.80e3T^{2} \)
59 \( 1 - 98.3iT - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 + 92iT - 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 56iT - 5.32e3T^{2} \)
79 \( 1 + 78T + 6.24e3T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 + 17.8iT - 7.92e3T^{2} \)
97 \( 1 + 32iT - 9.40e3T^{2} \)
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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

−15.20792972282815476015571279133, −13.68175108496696622664396584733, −12.96917461784470061224317786048, −11.48153338687204257967085112029, −10.05556224265148204549474634162, −8.850276241935608498957155139665, −8.347297233587131691794432032574, −5.94137585119197281268068358720, −4.62424247316482503301892295422, −2.58186362428924905480915729866, 2.17131723727890984097766088136, 4.08223328678377031679040300796, 6.70381367980156212657825159348, 7.18236054184151654360121377292, 8.880679047898305080751982526464, 10.12245583978878001683934168376, 11.30211096781844578266825975358, 12.87789159468169518770330368802, 13.78758732139391218205172937444, 14.47621702430565846663890279212

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