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Properties

Label 2-60-60.59-c5-0-1
Degree $2$
Conductor $60$
Sign $-0.526 + 0.850i$
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  + (−4.16 + 3.83i)2-s + (7.28 + 13.7i)3-s + (2.66 − 31.8i)4-s + (45.7 − 32.1i)5-s + (−83.1 − 29.4i)6-s − 213.·7-s + (111. + 142. i)8-s + (−136. + 200. i)9-s + (−67.0 + 309. i)10-s − 602.·11-s + (458. − 195. i)12-s − 333. i·13-s + (887. − 816. i)14-s + (776. + 395. i)15-s + (−1.00e3 − 169. i)16-s − 831.·17-s + ⋯
L(s)  = 1  + (−0.735 + 0.677i)2-s + (0.467 + 0.883i)3-s + (0.0831 − 0.996i)4-s + (0.817 − 0.575i)5-s + (−0.942 − 0.334i)6-s − 1.64·7-s + (0.613 + 0.789i)8-s + (−0.562 + 0.826i)9-s + (−0.212 + 0.977i)10-s − 1.50·11-s + (0.919 − 0.392i)12-s − 0.547i·13-s + (1.21 − 1.11i)14-s + (0.891 + 0.453i)15-s + (−0.986 − 0.165i)16-s − 0.697·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.526 + 0.850i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.526 + 0.850i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0242226 - 0.0434855i\)
\(L(\frac12)\) \(\approx\) \(0.0242226 - 0.0434855i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.16 - 3.83i)T \)
3 \( 1 + (-7.28 - 13.7i)T \)
5 \( 1 + (-45.7 + 32.1i)T \)
good7 \( 1 + 213.T + 1.68e4T^{2} \)
11 \( 1 + 602.T + 1.61e5T^{2} \)
13 \( 1 + 333. iT - 3.71e5T^{2} \)
17 \( 1 + 831.T + 1.41e6T^{2} \)
19 \( 1 + 1.07e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.99e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.59e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.39e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.66e3iT - 6.93e7T^{2} \)
41 \( 1 - 669. iT - 1.15e8T^{2} \)
43 \( 1 - 1.84e3T + 1.47e8T^{2} \)
47 \( 1 - 1.40e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.97e4T + 4.18e8T^{2} \)
59 \( 1 + 2.01e4T + 7.14e8T^{2} \)
61 \( 1 + 1.68e4T + 8.44e8T^{2} \)
67 \( 1 - 869.T + 1.35e9T^{2} \)
71 \( 1 - 3.58e4T + 1.80e9T^{2} \)
73 \( 1 + 3.71e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.09e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.86e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.98e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.69e5iT - 8.58e9T^{2} \)
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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

−15.46382944141264756269430322581, −13.66109513238477355475535282055, −13.11097893077267401797280142042, −10.74333684199976112552742284813, −9.805940943149041539148364655852, −9.245623001267070248113107215761, −7.916505079749715440766389713128, −6.16957649836348943151342725287, −5.03158872816675593643060875818, −2.69878818878038492903316119488, 0.02552222137596180779181465031, 2.20727872402665760464543948513, 3.17957542890016128758707103751, 6.29643192094736541464065204111, 7.26533905360882204648806926982, 8.773942641984740723352664435402, 9.819490990376009094209003525604, 10.77828429343061765395856391175, 12.54466145025891684615501344851, 13.02226874317569937087008621648

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