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Properties

Label 2-60-60.59-c5-0-47
Degree $2$
Conductor $60$
Sign $-0.819 + 0.573i$
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  + (5.44 − 1.52i)2-s + (−11.2 + 10.8i)3-s + (27.3 − 16.6i)4-s + (−42.6 − 36.1i)5-s + (−44.4 + 76.1i)6-s − 128.·7-s + (123. − 132. i)8-s + (8.22 − 242. i)9-s + (−287. − 132. i)10-s − 306.·11-s + (−126. + 482. i)12-s − 408. i·13-s + (−702. + 196. i)14-s + (869. − 55.9i)15-s + (470. − 909. i)16-s − 1.24e3·17-s + ⋯
L(s)  = 1  + (0.962 − 0.269i)2-s + (−0.718 + 0.695i)3-s + (0.854 − 0.519i)4-s + (−0.762 − 0.647i)5-s + (−0.504 + 0.863i)6-s − 0.994·7-s + (0.682 − 0.731i)8-s + (0.0338 − 0.999i)9-s + (−0.908 − 0.417i)10-s − 0.764·11-s + (−0.252 + 0.967i)12-s − 0.670i·13-s + (−0.957 + 0.268i)14-s + (0.997 − 0.0642i)15-s + (0.459 − 0.888i)16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.819 + 0.573i$
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: \((2,\ 60,\ (\ :5/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.264252 - 0.837822i\)
\(L(\frac12)\) \(\approx\) \(0.264252 - 0.837822i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.44 + 1.52i)T \)
3 \( 1 + (11.2 - 10.8i)T \)
5 \( 1 + (42.6 + 36.1i)T \)
good7 \( 1 + 128.T + 1.68e4T^{2} \)
11 \( 1 + 306.T + 1.61e5T^{2} \)
13 \( 1 + 408. iT - 3.71e5T^{2} \)
17 \( 1 + 1.24e3T + 1.41e6T^{2} \)
19 \( 1 + 1.54e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.93e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.47e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.18e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.16e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.56e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.90e4T + 1.47e8T^{2} \)
47 \( 1 + 1.15e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.36e4T + 4.18e8T^{2} \)
59 \( 1 - 1.93e4T + 7.14e8T^{2} \)
61 \( 1 - 3.41e4T + 8.44e8T^{2} \)
67 \( 1 + 4.93e4T + 1.35e9T^{2} \)
71 \( 1 + 7.17e4T + 1.80e9T^{2} \)
73 \( 1 + 8.10e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.16e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.59e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.76e3iT - 5.58e9T^{2} \)
97 \( 1 + 1.23e5iT - 8.58e9T^{2} \)
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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

−13.21389908671278843297113216206, −12.62407753832330857059539645812, −11.42924376980376194032234957851, −10.54576490989189531524046755013, −9.198133014822718203373037036491, −7.14486423913171324959858794201, −5.68164730660996074611117886421, −4.56801026263772810933124185642, −3.25005617778277645283098199852, −0.31291804168690676774094186129, 2.59565327527118696429321062397, 4.32215505138571742015465687187, 6.08288284648487039752654744084, 6.88754489071694443866143062580, 8.017668650392641955577178891986, 10.44544821418151074174029051642, 11.45651662845760585920704874083, 12.41924406618575171435349367622, 13.26762854916024668559716607311, 14.39339279203465635681585932179

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