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Properties

Label 2-60-5.4-c5-0-4
Degree $2$
Conductor $60$
Sign $0.225 + 0.974i$
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  − 9i·3-s + (54.4 − 12.5i)5-s − 12.5i·7-s − 81·9-s + 164.·11-s − 849. i·13-s + (−113. − 490. i)15-s − 1.60e3i·17-s + 446.·19-s − 113.·21-s − 802. i·23-s + (2.80e3 − 1.37e3i)25-s + 729i·27-s − 4.47e3·29-s + 5.20e3·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.974 − 0.225i)5-s − 0.0970i·7-s − 0.333·9-s + 0.410·11-s − 1.39i·13-s + (−0.130 − 0.562i)15-s − 1.34i·17-s + 0.283·19-s − 0.0560·21-s − 0.316i·23-s + (0.898 − 0.438i)25-s + 0.192i·27-s − 0.988·29-s + 0.972·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.49399 - 1.18809i\)
\(L(\frac12)\) \(\approx\) \(1.49399 - 1.18809i\)
\(L(\frac{7}{2})\) 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 \( 1 + 9iT \)
5 \( 1 + (-54.4 + 12.5i)T \)
good7 \( 1 + 12.5iT - 1.68e4T^{2} \)
11 \( 1 - 164.T + 1.61e5T^{2} \)
13 \( 1 + 849. iT - 3.71e5T^{2} \)
17 \( 1 + 1.60e3iT - 1.41e6T^{2} \)
19 \( 1 - 446.T + 2.47e6T^{2} \)
23 \( 1 + 802. iT - 6.43e6T^{2} \)
29 \( 1 + 4.47e3T + 2.05e7T^{2} \)
31 \( 1 - 5.20e3T + 2.86e7T^{2} \)
37 \( 1 - 8.24e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.25e3T + 1.15e8T^{2} \)
43 \( 1 - 1.13e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.98e4iT - 2.29e8T^{2} \)
53 \( 1 + 9.19e3iT - 4.18e8T^{2} \)
59 \( 1 + 9.94e3T + 7.14e8T^{2} \)
61 \( 1 - 4.15e4T + 8.44e8T^{2} \)
67 \( 1 - 4.90e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.49e4T + 1.80e9T^{2} \)
73 \( 1 + 6.56e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.71e4T + 3.07e9T^{2} \)
83 \( 1 + 5.48e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.39e4T + 5.58e9T^{2} \)
97 \( 1 - 1.00e5iT - 8.58e9T^{2} \)
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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

−13.67349717042440105017714315219, −12.92348426402861398255258363917, −11.71277299143022740469636742483, −10.28664893004472552184324715701, −9.171920249696155328090628214360, −7.76558099031349814207925433357, −6.36765031417523218956962420017, −5.12935293915114756945656540222, −2.78195457348448520337036897411, −0.982253561159099034486744916402, 1.92885105946681168061283939816, 3.91534317820813561339966048519, 5.57423476938506258265881295382, 6.79660940436603963433598182247, 8.733008028962531419928126889387, 9.673430331856047187993036637152, 10.74291460740430297302713676432, 11.97314926832914722816322889295, 13.43456930646891361783513431760, 14.33258491634404989069136222635

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