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Properties

Label 2-60-15.8-c5-0-8
Degree $2$
Conductor $60$
Sign $-0.262 + 0.964i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.90 − 12.0i)3-s + (51.4 + 21.7i)5-s + (56.0 − 56.0i)7-s + (−46.9 + 238. i)9-s − 616. i·11-s + (−323. − 323. i)13-s + (−247. − 835. i)15-s + (−1.37e3 − 1.37e3i)17-s + 256. i·19-s + (−1.22e3 − 119. i)21-s + (2.64e3 − 2.64e3i)23-s + (2.17e3 + 2.24e3i)25-s + (3.33e3 − 1.79e3i)27-s + 381.·29-s − 7.35e3·31-s + ⋯
L(s)  = 1  + (−0.635 − 0.772i)3-s + (0.921 + 0.389i)5-s + (0.432 − 0.432i)7-s + (−0.193 + 0.981i)9-s − 1.53i·11-s + (−0.531 − 0.531i)13-s + (−0.284 − 0.958i)15-s + (−1.15 − 1.15i)17-s + 0.162i·19-s + (−0.608 − 0.0592i)21-s + (1.04 − 1.04i)23-s + (0.696 + 0.717i)25-s + (0.880 − 0.474i)27-s + 0.0841·29-s − 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.262 + 0.964i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.262 + 0.964i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.838529 - 1.09699i\)
\(L(\frac12)\) \(\approx\) \(0.838529 - 1.09699i\)
\(L(\frac{7}{2})\) 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 \( 1 + (9.90 + 12.0i)T \)
5 \( 1 + (-51.4 - 21.7i)T \)
good7 \( 1 + (-56.0 + 56.0i)T - 1.68e4iT^{2} \)
11 \( 1 + 616. iT - 1.61e5T^{2} \)
13 \( 1 + (323. + 323. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.37e3 + 1.37e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 256. iT - 2.47e6T^{2} \)
23 \( 1 + (-2.64e3 + 2.64e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 381.T + 2.05e7T^{2} \)
31 \( 1 + 7.35e3T + 2.86e7T^{2} \)
37 \( 1 + (-6.06e3 + 6.06e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.46e3iT - 1.15e8T^{2} \)
43 \( 1 + (-3.45e3 - 3.45e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-6.96e3 - 6.96e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-5.26e3 + 5.26e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 3.38e4T + 7.14e8T^{2} \)
61 \( 1 + 3.11e4T + 8.44e8T^{2} \)
67 \( 1 + (1.57e4 - 1.57e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.59e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.10e4 - 5.10e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 4.20e4iT - 3.07e9T^{2} \)
83 \( 1 + (-4.89e4 + 4.89e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 2.18e4T + 5.58e9T^{2} \)
97 \( 1 + (-6.35e4 + 6.35e4i)T - 8.58e9iT^{2} \)
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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

−13.66183814172105663937904896535, −12.83568698769381780069093100805, −11.23759137675409185450206106673, −10.70429215543612106782426920711, −8.984817093978508713754479759514, −7.45646586615752279345374178915, −6.29968308406348399535448388623, −5.12655857718198952531502663829, −2.57005419486072471810458502607, −0.71501107966253410070580163007, 1.91589445989930741745651502686, 4.43170614412232638211911516638, 5.43168548910998560881752348953, 6.86351247016963620435517045911, 8.942346483552574969488970525849, 9.732405404169390495968165701036, 10.89479369926938437189183633796, 12.13781584146912230193590454141, 13.13435868256133038324239153646, 14.76157976018952743785661558802

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