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Properties

Label 2-60-5.2-c2-0-1
Degree $2$
Conductor $60$
Sign $0.828 + 0.560i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (1.77 − 4.67i)5-s + (2.55 + 2.55i)7-s − 2.99i·9-s + 8.24·11-s + (−12.2 + 12.2i)13-s + (−3.55 − 7.89i)15-s + (−12.4 − 12.4i)17-s + 34.4i·19-s + 6.24·21-s + (−17.3 + 17.3i)23-s + (−18.6 − 16.5i)25-s + (−3.67 − 3.67i)27-s − 9.75i·29-s + 28.4·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.355 − 0.934i)5-s + (0.364 + 0.364i)7-s − 0.333i·9-s + 0.749·11-s + (−0.942 + 0.942i)13-s + (−0.236 − 0.526i)15-s + (−0.732 − 0.732i)17-s + 1.81i·19-s + 0.297·21-s + (−0.754 + 0.754i)23-s + (−0.747 − 0.663i)25-s + (−0.136 − 0.136i)27-s − 0.336i·29-s + 0.919·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.828 + 0.560i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32362 - 0.405671i\)
\(L(\frac12)\) \(\approx\) \(1.32362 - 0.405671i\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-1.77 + 4.67i)T \)
good7 \( 1 + (-2.55 - 2.55i)T + 49iT^{2} \)
11 \( 1 - 8.24T + 121T^{2} \)
13 \( 1 + (12.2 - 12.2i)T - 169iT^{2} \)
17 \( 1 + (12.4 + 12.4i)T + 289iT^{2} \)
19 \( 1 - 34.4iT - 361T^{2} \)
23 \( 1 + (17.3 - 17.3i)T - 529iT^{2} \)
29 \( 1 + 9.75iT - 841T^{2} \)
31 \( 1 - 28.4T + 961T^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 - 74.4T + 1.68e3T^{2} \)
43 \( 1 + (-34.8 + 34.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (22.0 + 22.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (64.6 - 64.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 15.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.5T + 3.72e3T^{2} \)
67 \( 1 + (-4.69 - 4.69i)T + 4.48e3iT^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 + (-34.1 + 34.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 0.494iT - 6.24e3T^{2} \)
83 \( 1 + (18.3 - 18.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 136. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.5 - 94.5i)T + 9.40e3iT^{2} \)
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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

−14.44409654507072317408591018221, −13.78719639931969976875329601789, −12.37408721763977182113714701285, −11.76302175829016663840506596213, −9.752001442517049992895065502273, −8.901707025285830534209501839058, −7.63768400789077808601818569442, −6.00781343260534877789931637244, −4.37152550539673925958012979198, −1.88682070467486011641506462672, 2.69932969823620307452100736355, 4.51166833113304230266642649236, 6.41310219126849547825542832129, 7.72911778925758555039472172740, 9.232000355991930346211153698283, 10.38667621646406219879299925245, 11.25947737135068778355463280302, 12.88941880554577659660561955972, 14.12703125270444538268536049172, 14.80819418895992646582293371954

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