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Properties

Label 2-60-4.3-c4-0-4
Degree $2$
Conductor $60$
Sign $-0.955 - 0.295i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.37 + 3.21i)2-s + 5.19i·3-s + (−4.72 + 15.2i)4-s − 11.1·5-s + (−16.7 + 12.3i)6-s + 19.2i·7-s + (−60.4 + 21.1i)8-s − 27·9-s + (−26.5 − 35.9i)10-s + 28.1i·11-s + (−79.4 − 24.5i)12-s − 28.6·13-s + (−62.0 + 45.8i)14-s − 58.0i·15-s + (−211. − 144. i)16-s + 290.·17-s + ⋯
L(s)  = 1  + (0.593 + 0.804i)2-s + 0.577i·3-s + (−0.295 + 0.955i)4-s − 0.447·5-s + (−0.464 + 0.342i)6-s + 0.393i·7-s + (−0.944 + 0.329i)8-s − 0.333·9-s + (−0.265 − 0.359i)10-s + 0.232i·11-s + (−0.551 − 0.170i)12-s − 0.169·13-s + (−0.316 + 0.233i)14-s − 0.258i·15-s + (−0.825 − 0.563i)16-s + 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.955 - 0.295i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.955 - 0.295i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.244761 + 1.62233i\)
\(L(\frac12)\) \(\approx\) \(0.244761 + 1.62233i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.37 - 3.21i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 + 11.1T \)
good7 \( 1 - 19.2iT - 2.40e3T^{2} \)
11 \( 1 - 28.1iT - 1.46e4T^{2} \)
13 \( 1 + 28.6T + 2.85e4T^{2} \)
17 \( 1 - 290.T + 8.35e4T^{2} \)
19 \( 1 - 459. iT - 1.30e5T^{2} \)
23 \( 1 - 63.0iT - 2.79e5T^{2} \)
29 \( 1 - 1.14e3T + 7.07e5T^{2} \)
31 \( 1 - 1.33e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.36e3T + 1.87e6T^{2} \)
41 \( 1 - 1.24e3T + 2.82e6T^{2} \)
43 \( 1 - 663. iT - 3.41e6T^{2} \)
47 \( 1 + 3.93e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.61e3T + 7.89e6T^{2} \)
59 \( 1 + 2.60e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.39e3T + 1.38e7T^{2} \)
67 \( 1 - 6.03e3iT - 2.01e7T^{2} \)
71 \( 1 + 955. iT - 2.54e7T^{2} \)
73 \( 1 + 8.81e3T + 2.83e7T^{2} \)
79 \( 1 + 3.71e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.99e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.08e4T + 6.27e7T^{2} \)
97 \( 1 - 5.58e3T + 8.85e7T^{2} \)
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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

−14.83168968308481067971338572211, −14.12906733573135042509267338026, −12.56745126819004366638888731960, −11.80287636427227073447270030956, −10.13363088907970852232269282349, −8.699797574441286954873605234796, −7.60164197888588060809380860398, −6.01330699600758947638664581584, −4.71581029973546148745158270715, −3.28766790833406107853764849688, 0.805737709779595408110730939730, 2.86203591334474061062358693691, 4.52851524806709164632418838203, 6.16745815721302866429080736906, 7.69752918331825215169044785235, 9.306453497732626658975619565775, 10.70639609405372772361788699486, 11.69037474461989558952749575865, 12.66292745245284780641617571216, 13.65834750065410118385017524177

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