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Properties

Label 2-60-12.11-c1-0-7
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  + (−0.599 − 1.28i)2-s + (−0.468 − 1.66i)3-s + (−1.28 + 1.53i)4-s − i·5-s + (−1.85 + 1.59i)6-s − 0.936i·7-s + (2.73 + 0.719i)8-s + (−2.56 + 1.56i)9-s + (−1.28 + 0.599i)10-s + 4.27·11-s + (3.16 + 1.41i)12-s + 3.12·13-s + (−1.19 + 0.561i)14-s + (−1.66 + 0.468i)15-s + (−0.719 − 3.93i)16-s + 2i·17-s + ⋯
L(s)  = 1  + (−0.424 − 0.905i)2-s + (−0.270 − 0.962i)3-s + (−0.640 + 0.768i)4-s − 0.447i·5-s + (−0.757 + 0.653i)6-s − 0.353i·7-s + (0.967 + 0.254i)8-s + (−0.853 + 0.520i)9-s + (−0.405 + 0.189i)10-s + 1.28·11-s + (0.912 + 0.408i)12-s + 0.866·13-s + (−0.320 + 0.150i)14-s + (−0.430 + 0.120i)15-s + (−0.179 − 0.983i)16-s + 0.485i·17-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: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358661 - 0.553742i\)
\(L(\frac12)\) \(\approx\) \(0.358661 - 0.553742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 + 1.28i)T \)
3 \( 1 + (0.468 + 1.66i)T \)
5 \( 1 + iT \)
good7 \( 1 + 0.936iT - 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4.27iT - 19T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 - 0.936T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 + 5.20iT - 67T^{2} \)
71 \( 1 + 6.67T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−14.22124995671631807239277133705, −13.41465179370277268212018520673, −12.26328448249659111929447292060, −11.61970529823996132447548400944, −10.28968930947388791546973913200, −8.817309967367604010386188190048, −7.82017111403550394787877621486, −6.15156372983368566590443624338, −3.95511662682755168969942725641, −1.51997553530814570175350957285, 4.03249840269348818970167844308, 5.66466421378226444009487880411, 6.79065912643316171426594024459, 8.617521518016947707233342324753, 9.465714620815556994031939318846, 10.66295884337075361557817200484, 11.79001083729951559403818715147, 13.80109034533139628445628266329, 14.63185158745170906865071408912, 15.62050146349201357222354456870

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