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Properties

Label 2-60-60.59-c5-0-54
Degree $2$
Conductor $60$
Sign $-0.663 - 0.748i$
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  + (3.08 − 4.73i)2-s + (−14.4 − 5.82i)3-s + (−12.9 − 29.2i)4-s + (43.0 − 35.7i)5-s + (−72.2 + 50.5i)6-s − 168.·7-s + (−178. − 29.2i)8-s + (175. + 168. i)9-s + (−36.3 − 314. i)10-s + 165.·11-s + (16.1 + 498. i)12-s + 182. i·13-s + (−519. + 796. i)14-s + (−829. + 265. i)15-s + (−690. + 756. i)16-s − 1.80e3·17-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.927 − 0.373i)3-s + (−0.403 − 0.914i)4-s + (0.769 − 0.638i)5-s + (−0.819 + 0.572i)6-s − 1.29·7-s + (−0.986 − 0.161i)8-s + (0.720 + 0.693i)9-s + (−0.114 − 0.993i)10-s + 0.412·11-s + (0.0323 + 0.999i)12-s + 0.299i·13-s + (−0.708 + 1.08i)14-s + (−0.952 + 0.304i)15-s + (−0.674 + 0.738i)16-s − 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.663 - 0.748i$
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: \((2,\ 60,\ (\ :5/2),\ -0.663 - 0.748i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.296891 + 0.659862i\)
\(L(\frac12)\) \(\approx\) \(0.296891 + 0.659862i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.08 + 4.73i)T \)
3 \( 1 + (14.4 + 5.82i)T \)
5 \( 1 + (-43.0 + 35.7i)T \)
good7 \( 1 + 168.T + 1.68e4T^{2} \)
11 \( 1 - 165.T + 1.61e5T^{2} \)
13 \( 1 - 182. iT - 3.71e5T^{2} \)
17 \( 1 + 1.80e3T + 1.41e6T^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + 391. iT - 6.43e6T^{2} \)
29 \( 1 - 382. iT - 2.05e7T^{2} \)
31 \( 1 + 8.73e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.38e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.35e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 - 1.60e4iT - 2.29e8T^{2} \)
53 \( 1 + 8.22e3T + 4.18e8T^{2} \)
59 \( 1 + 7.47e3T + 7.14e8T^{2} \)
61 \( 1 + 1.23e4T + 8.44e8T^{2} \)
67 \( 1 - 6.06e4T + 1.35e9T^{2} \)
71 \( 1 + 2.83e4T + 1.80e9T^{2} \)
73 \( 1 - 4.02e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.46e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.54e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.57e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.34e5iT - 8.58e9T^{2} \)
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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

−13.01975542585498240419758365412, −12.49155481427070119871315441301, −11.24781669717018261958531082350, −10.04584240302353200261081594280, −9.137941565703894571942117238972, −6.57680992752343659050865857098, −5.74954627693926024211554808256, −4.23432903446890211887728572028, −2.02525444238102177711289574483, −0.30803295985869228752731164199, 3.24861874515910702519810878182, 4.95078746614051375280206378106, 6.46198313885120774415113558728, 6.70345517623677096729622386924, 9.067110424454709663180790772882, 10.12538313944411034467721311169, 11.50991809481803628694664589603, 12.86796567092330895302836908279, 13.57399370452378824819195587134, 15.05856265709817019670008233973

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