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Properties

Label 2-60-15.2-c5-0-3
Degree $2$
Conductor $60$
Sign $0.778 - 0.627i$
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  + (15.5 − 0.876i)3-s + (−55.7 + 3.84i)5-s + (151. + 151. i)7-s + (241. − 27.2i)9-s + 504. i·11-s + (567. − 567. i)13-s + (−864. + 108. i)15-s + (91.1 − 91.1i)17-s + 600. i·19-s + (2.48e3 + 2.22e3i)21-s + (2.15e3 + 2.15e3i)23-s + (3.09e3 − 429. i)25-s + (3.73e3 − 636. i)27-s − 5.81e3·29-s − 2.22e3·31-s + ⋯
L(s)  = 1  + (0.998 − 0.0562i)3-s + (−0.997 + 0.0688i)5-s + (1.16 + 1.16i)7-s + (0.993 − 0.112i)9-s + 1.25i·11-s + (0.931 − 0.931i)13-s + (−0.992 + 0.124i)15-s + (0.0765 − 0.0765i)17-s + 0.381i·19-s + (1.23 + 1.09i)21-s + (0.850 + 0.850i)23-s + (0.990 − 0.137i)25-s + (0.985 − 0.167i)27-s − 1.28·29-s − 0.414·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.17477 + 0.767693i\)
\(L(\frac12)\) \(\approx\) \(2.17477 + 0.767693i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.5 + 0.876i)T \)
5 \( 1 + (55.7 - 3.84i)T \)
good7 \( 1 + (-151. - 151. i)T + 1.68e4iT^{2} \)
11 \( 1 - 504. iT - 1.61e5T^{2} \)
13 \( 1 + (-567. + 567. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-91.1 + 91.1i)T - 1.41e6iT^{2} \)
19 \( 1 - 600. iT - 2.47e6T^{2} \)
23 \( 1 + (-2.15e3 - 2.15e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 5.81e3T + 2.05e7T^{2} \)
31 \( 1 + 2.22e3T + 2.86e7T^{2} \)
37 \( 1 + (6.79e3 + 6.79e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.21e4iT - 1.15e8T^{2} \)
43 \( 1 + (5.94e3 - 5.94e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-4.50e3 + 4.50e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (2.60e4 + 2.60e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 6.53e3T + 7.14e8T^{2} \)
61 \( 1 - 2.84e4T + 8.44e8T^{2} \)
67 \( 1 + (1.23e4 + 1.23e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.92e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.82e4 - 1.82e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 4.11e4iT - 3.07e9T^{2} \)
83 \( 1 + (-5.60e4 - 5.60e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 5.46e4T + 5.58e9T^{2} \)
97 \( 1 + (-1.70e4 - 1.70e4i)T + 8.58e9iT^{2} \)
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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

−14.61757287845056777902423115949, −13.00388436137283671896679039894, −12.06321370734081764014870991127, −10.86532543735400152372190299256, −9.204473150383760969582085657210, −8.209928268020277692429226465847, −7.37460534375604725718021547439, −5.13582888532454738106461004440, −3.57100722803237230125024593531, −1.84366916428046105512288019589, 1.18159219625583356475516899843, 3.50082044819433409079062971779, 4.52390094109310185850009906476, 7.00858316689185190639077504009, 8.106262444003484266918355018024, 8.860224918351791964744481395174, 10.76315314617168104230106361395, 11.41406033469542104962644964791, 13.18083088043317518003543884232, 14.05045197249660713230832754583

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