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Properties

Label 2-60-5.2-c6-0-5
Degree $2$
Conductor $60$
Sign $-0.173 + 0.984i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.0 − 11.0i)3-s + (124. − 7.22i)5-s + (−434. − 434. i)7-s − 242. i·9-s + 1.74e3·11-s + (−1.54e3 + 1.54e3i)13-s + (1.29e3 − 1.45e3i)15-s + (−4.01e3 − 4.01e3i)17-s − 1.32e4i·19-s − 9.58e3·21-s + (−104. + 104. i)23-s + (1.55e4 − 1.80e3i)25-s + (−2.67e3 − 2.67e3i)27-s − 2.56e4i·29-s + 2.11e3·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.998 − 0.0577i)5-s + (−1.26 − 1.26i)7-s − 0.333i·9-s + 1.30·11-s + (−0.703 + 0.703i)13-s + (0.383 − 0.431i)15-s + (−0.816 − 0.816i)17-s − 1.92i·19-s − 1.03·21-s + (−0.00857 + 0.00857i)23-s + (0.993 − 0.115i)25-s + (−0.136 − 0.136i)27-s − 1.05i·29-s + 0.0710·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.24165 - 1.47894i\)
\(L(\frac12)\) \(\approx\) \(1.24165 - 1.47894i\)
\(L(4)\) 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 \( 1 + (-11.0 + 11.0i)T \)
5 \( 1 + (-124. + 7.22i)T \)
good7 \( 1 + (434. + 434. i)T + 1.17e5iT^{2} \)
11 \( 1 - 1.74e3T + 1.77e6T^{2} \)
13 \( 1 + (1.54e3 - 1.54e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (4.01e3 + 4.01e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 1.32e4iT - 4.70e7T^{2} \)
23 \( 1 + (104. - 104. i)T - 1.48e8iT^{2} \)
29 \( 1 + 2.56e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.11e3T + 8.87e8T^{2} \)
37 \( 1 + (-7.05e3 - 7.05e3i)T + 2.56e9iT^{2} \)
41 \( 1 - 5.99e3T + 4.75e9T^{2} \)
43 \( 1 + (4.97e3 - 4.97e3i)T - 6.32e9iT^{2} \)
47 \( 1 + (-1.33e5 - 1.33e5i)T + 1.07e10iT^{2} \)
53 \( 1 + (1.63e5 - 1.63e5i)T - 2.21e10iT^{2} \)
59 \( 1 + 2.42e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.90e5T + 5.15e10T^{2} \)
67 \( 1 + (-2.82e5 - 2.82e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 3.40e5T + 1.28e11T^{2} \)
73 \( 1 + (7.63e4 - 7.63e4i)T - 1.51e11iT^{2} \)
79 \( 1 - 5.32e5iT - 2.43e11T^{2} \)
83 \( 1 + (7.17e4 - 7.17e4i)T - 3.26e11iT^{2} \)
89 \( 1 - 1.96e5iT - 4.96e11T^{2} \)
97 \( 1 + (9.46e4 + 9.46e4i)T + 8.32e11iT^{2} \)
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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

−13.65884192701192730822248269219, −12.75321970877980796443144418332, −11.23139245644457134324748373793, −9.636687841314522538462734055023, −9.234023391313646544234824050114, −7.02069597139899362215629566343, −6.54492622576398567734615108390, −4.33592713397161723848058969418, −2.59958507816849087061340644206, −0.75823306555082617690184969821, 2.06282592219703102641813443883, 3.50635428524111367943529006243, 5.58512044679578212697655900495, 6.53818984292827401633096870623, 8.635928604025314222054230225150, 9.494690769731410270652906121378, 10.32205685640848334634027531376, 12.18573062369532530997029845006, 12.94962499284870577051962306363, 14.34447002981409527999628037289

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