↑

Properties

Label 2-60-15.2-c1-0-0
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

Related objects

Downloads

Learn more

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: $\chi_{60} (17, \cdot )$
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 \( 1 + (-9 - 9i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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