Properties

Label 2-60-60.59-c1-0-6
Degree $2$
Conductor $60$
Sign $0.408 + 0.912i$
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  − 1.41i·2-s + (1.58 − 0.707i)3-s − 2.00·4-s + 2.23i·5-s + (−1.00 − 2.23i)6-s − 3.16·7-s + 2.82i·8-s + (2.00 − 2.23i)9-s + 3.16·10-s + (−3.16 + 1.41i)12-s + 4.47i·14-s + (1.58 + 3.53i)15-s + 4.00·16-s + (−3.16 − 2.82i)18-s − 4.47i·20-s + (−5.00 + 2.23i)21-s + ⋯
L(s)  = 1  − 0.999i·2-s + (0.912 − 0.408i)3-s − 1.00·4-s + 0.999i·5-s + (−0.408 − 0.912i)6-s − 1.19·7-s + 1.00i·8-s + (0.666 − 0.745i)9-s + 1.00·10-s + (−0.912 + 0.408i)12-s + 1.19i·14-s + (0.408 + 0.912i)15-s + 1.00·16-s + (−0.745 − 0.666i)18-s − 1.00i·20-s + (−1.09 + 0.487i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801279 - 0.519414i\)
\(L(\frac12)\) \(\approx\) \(0.801279 - 0.519414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-1.58 + 0.707i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 8.94iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 - 9.89iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 15.8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 - 17.8iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.57174709672473076552998765338, −13.60581563496821314175196533636, −12.81931457903855553437122451891, −11.56791799856056849947225330787, −10.12134331194087099550001736512, −9.410288330450700677374442105804, −7.88405029260432313185638045755, −6.39099582234643412308893032837, −3.73926876494475095547252880784, −2.60713254643978157043709673537, 3.71190222307801709910414989482, 5.20614278960446912379418776037, 6.91635656833683718169918462311, 8.362570555051921221577842780062, 9.190335436100818323413861275335, 10.11107562427893349004277655018, 12.58080510332568281047636415465, 13.25849038978898880128775411339, 14.31136650228818586631341169721, 15.52622523045204036223128127746

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