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Properties

Label 2-60-60.59-c5-0-46
Degree $2$
Conductor $60$
Sign $0.302 + 0.953i$
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  + (5.59 − 0.866i)2-s − 15.5i·3-s + (30.5 − 9.68i)4-s + 55.9·5-s + (−13.5 − 87.1i)6-s + (162. − 80.5i)8-s − 243·9-s + (312.5 − 48.4i)10-s + (−150. − 475. i)12-s − 871. i·15-s + (836. − 590. i)16-s − 648.·17-s + (−1.35e3 + 210. i)18-s − 2.28e3i·19-s + (1.70e3 − 541. i)20-s + ⋯
L(s)  = 1  + (0.988 − 0.153i)2-s − 0.999i·3-s + (0.953 − 0.302i)4-s + 0.999·5-s + (−0.153 − 0.988i)6-s + (0.895 − 0.444i)8-s − 9-s + (0.988 − 0.153i)10-s + (−0.302 − 0.953i)12-s − 0.999i·15-s + (0.816 − 0.576i)16-s − 0.544·17-s + (−0.988 + 0.153i)18-s − 1.45i·19-s + (0.953 − 0.302i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.302 + 0.953i$
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: \((2,\ 60,\ (\ :5/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.81480 - 2.05965i\)
\(L(\frac12)\) \(\approx\) \(2.81480 - 2.05965i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.59 + 0.866i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 + 648.T + 1.41e6T^{2} \)
19 \( 1 + 2.28e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.88e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.05e7T^{2} \)
31 \( 1 - 6.93e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 - 1.15e8T^{2} \)
43 \( 1 + 1.47e8T^{2} \)
47 \( 1 - 2.77e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.08e4T + 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 3.48e4T + 8.44e8T^{2} \)
67 \( 1 + 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 8.60e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.02e5iT - 3.93e9T^{2} \)
89 \( 1 - 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−13.54454720558695906477139707123, −13.11239558815593583217437899951, −11.84602483174404413404901511751, −10.79724516953266424223144746215, −9.174704404545756585303668468812, −7.32911190645733895662670404555, −6.29756211969504577696191430782, −5.13291908889729115953599852162, −2.86431614034986513348992062963, −1.49171985602151090481865763173, 2.40896881552571179828165298562, 4.07811486877285194194074388872, 5.39390020442934926710204035211, 6.44202225634761342080298676913, 8.425147169944319396014685827529, 9.959543318701449538212728505803, 10.82609785749208280872987466956, 12.18596915138162414373245990680, 13.42828577820956007819758713621, 14.40968672498824131132511162079

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