Properties

Label 2-60-60.59-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  − 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

\(L(1)\) \(\approx\) \(0.374372 - 0.577529i\)
\(L(\frac12)\) \(\approx\) \(0.374372 - 0.577529i\)
\(L(\frac{3}{2})\) 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

−14.40884061334837374941686196439, −13.23077157856824661009931669758, −12.32212370350680719602150609768, −11.49501306157240950308715152306, −10.52909702441978037345575630526, −8.988021565473084263259272139074, −7.76814223664752297023615196726, −5.47125061701447773016299072657, −4.52760392352707541403596359059, −1.48983611877484028104485383038, 4.24345971898683301440855894128, 5.61045984548333734292758686337, 6.81832180906283164271842194157, 8.057232819118948939437920310797, 9.747336975782428837565256190701, 10.86350805723286984809499761141, 11.93361323158611850294817609907, 13.58696131546888214264784967849, 14.75183761004493719035284875953, 15.31526433221270455487829857567

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