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Properties

Label 2-60-20.19-c2-0-5
Degree $2$
Conductor $60$
Sign $0.866 + 0.5i$
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  + (−1.73 + i)2-s − 1.73·3-s + (1.99 − 3.46i)4-s − 5i·5-s + (2.99 − 1.73i)6-s + 10.3·7-s + 7.99i·8-s + 2.99·9-s + (5 + 8.66i)10-s − 10.3i·11-s + (−3.46 + 5.99i)12-s − 18i·13-s + (−18 + 10.3i)14-s + 8.66i·15-s + (−8 − 13.8i)16-s + 10i·17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s − 0.577·3-s + (0.499 − 0.866i)4-s − i·5-s + (0.499 − 0.288i)6-s + 1.48·7-s + 0.999i·8-s + 0.333·9-s + (0.5 + 0.866i)10-s − 0.944i·11-s + (−0.288 + 0.499i)12-s − 1.38i·13-s + (−1.28 + 0.742i)14-s + 0.577i·15-s + (−0.5 − 0.866i)16-s + 0.588i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.729700 - 0.195522i\)
\(L(\frac12)\) \(\approx\) \(0.729700 - 0.195522i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - i)T \)
3 \( 1 + 1.73T \)
5 \( 1 + 5iT \)
good7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 + 10.3iT - 121T^{2} \)
13 \( 1 + 18iT - 169T^{2} \)
17 \( 1 - 10iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + 6.92T + 529T^{2} \)
29 \( 1 - 36T + 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 - 54iT - 1.36e3T^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 + 20.7T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 26iT - 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 41.5T + 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + 36iT - 5.32e3T^{2} \)
79 \( 1 - 90.0iT - 6.24e3T^{2} \)
83 \( 1 - 90.0T + 6.88e3T^{2} \)
89 \( 1 - 18T + 7.92e3T^{2} \)
97 \( 1 + 72iT - 9.40e3T^{2} \)
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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

−15.07108089970204556871019808388, −13.81794256866553787585069732694, −12.24142241618246264743057594057, −11.15166645382640786618117830741, −10.15919050995604591110642248119, −8.436611060994473593840948569726, −7.992975152404441093536453226864, −5.94485614953169431329086413970, −4.95892377775252518668236308255, −1.16514509826015671606039577509, 2.06380075067946740643301729698, 4.47763910781306881374753849106, 6.74629279012048726898290484259, 7.68290578383106784099976878262, 9.277973850975125519896226242034, 10.55420215064691143691321790971, 11.40076212512292228537073202896, 12.06677220087979715351653774115, 13.89723491415712413328517360833, 15.02428783842420192773428730720

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