Properties

Label 2-60-15.8-c1-0-1
Degree $2$
Conductor $60$
Sign $0.982 + 0.187i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 0.618i)3-s − 2.23·5-s + (−1 + i)7-s + (2.23 − 2.00i)9-s + 4.47i·11-s + (−3 − 3i)13-s + (−3.61 + 1.38i)15-s + (−2.23 − 2.23i)17-s + 2i·19-s + (−1 + 2.23i)21-s + (2.23 − 2.23i)23-s + 5.00·25-s + (2.38 − 4.61i)27-s + 4.47·29-s + 4·31-s + ⋯
L(s)  = 1  + (0.934 − 0.356i)3-s − 0.999·5-s + (−0.377 + 0.377i)7-s + (0.745 − 0.666i)9-s + 1.34i·11-s + (−0.832 − 0.832i)13-s + (−0.934 + 0.356i)15-s + (−0.542 − 0.542i)17-s + 0.458i·19-s + (−0.218 + 0.487i)21-s + (0.466 − 0.466i)23-s + 1.00·25-s + (0.458 − 0.888i)27-s + 0.830·29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.982 + 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961907 - 0.0910313i\)
\(L(\frac12)\) \(\approx\) \(0.961907 - 0.0910313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.618i)T \)
5 \( 1 + 2.23T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (2.23 + 2.23i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.23 + 2.23i)T - 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (-6.70 - 6.70i)T + 47iT^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (6.70 - 6.70i)T - 83iT^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (-9 + 9i)T - 97iT^{2} \)
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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

−15.28500223732656605554352204650, −14.09109952457105515098178140454, −12.58402950472067078764952571400, −12.19574659536236122770053020907, −10.30783413356490826296638639362, −9.091176975443855831070908195766, −7.84264132045831719729779735590, −6.92382608257770560132859887593, −4.54704649572272101623991758334, −2.80493384870982070917755980699, 3.18642659840942046056459666008, 4.50603179780288941406115005502, 6.86445969136353655626133935062, 8.141539412500296434633534646016, 9.122024583398483955409754999430, 10.53741246784563827321850000183, 11.68717685213666173867539920033, 13.14322939861203385695080869198, 14.09384294321290039682891319342, 15.17529991858808094220647482887

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