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Properties

Label 2-60-3.2-c2-0-0
Degree $2$
Conductor $60$
Sign $0.745 - 0.666i$
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 + 2.23i)3-s + 2.23i·5-s + 2·7-s + (−1.00 + 8.94i)9-s − 13.4i·11-s + 8·13-s + (−5.00 + 4.47i)15-s − 13.4i·17-s − 34·19-s + (4 + 4.47i)21-s − 40.2i·23-s − 5.00·25-s + (−22.0 + 15.6i)27-s + 40.2i·29-s + 14·31-s + ⋯
L(s)  = 1  + (0.666 + 0.745i)3-s + 0.447i·5-s + 0.285·7-s + (−0.111 + 0.993i)9-s − 1.21i·11-s + 0.615·13-s + (−0.333 + 0.298i)15-s − 0.789i·17-s − 1.78·19-s + (0.190 + 0.212i)21-s − 1.74i·23-s − 0.200·25-s + (−0.814 + 0.579i)27-s + 1.38i·29-s + 0.451·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.745 - 0.666i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32431 + 0.505842i\)
\(L(\frac12)\) \(\approx\) \(1.32431 + 0.505842i\)
\(L(2)\) 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 \( 1 + (-2 - 2.23i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 - 2T + 49T^{2} \)
11 \( 1 + 13.4iT - 121T^{2} \)
13 \( 1 - 8T + 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 + 34T + 361T^{2} \)
23 \( 1 + 40.2iT - 529T^{2} \)
29 \( 1 - 40.2iT - 841T^{2} \)
31 \( 1 - 14T + 961T^{2} \)
37 \( 1 - 56T + 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 - 8T + 1.84e3T^{2} \)
47 \( 1 - 40.2iT - 2.20e3T^{2} \)
53 \( 1 - 40.2iT - 2.80e3T^{2} \)
59 \( 1 + 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 46T + 3.72e3T^{2} \)
67 \( 1 - 32T + 4.48e3T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 106T + 5.32e3T^{2} \)
79 \( 1 + 22T + 6.24e3T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 + 107. iT - 7.92e3T^{2} \)
97 \( 1 - 122T + 9.40e3T^{2} \)
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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

−14.79968700639781218625433619833, −14.15359525255581184575442336732, −12.98218510276332119757938288328, −11.16421606278365258658772828018, −10.53223108714509347242375660529, −8.985168264332896447058075911239, −8.127246269305760822398265053348, −6.29280108209387160444954218493, −4.47219991278825650452245220563, −2.87606472706765793080168502350, 1.89223480576888286366263656030, 4.14352290029282797770081817173, 6.18085250429145341658873508604, 7.63103121951426138730849172186, 8.631354096503470311805115270511, 9.883757467932475121419756034879, 11.55079898514535797934876780922, 12.73072319139510669475686442289, 13.40238793048262111745947212203, 14.77466625208214471714713709051

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