Properties

Label 2-60-5.4-c7-0-1
Degree $2$
Conductor $60$
Sign $0.447 - 0.894i$
Analytic cond. $18.7431$
Root an. cond. $4.32933$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27i·3-s + (250 + 125i)5-s + 722i·7-s − 729·9-s − 3.99e3·11-s + 3.03e3i·13-s + (3.37e3 − 6.75e3i)15-s + 2.05e4i·17-s + 2.53e4·19-s + 1.94e4·21-s + 6.66e4i·23-s + (4.68e4 + 6.25e4i)25-s + 1.96e4i·27-s + 1.52e5·29-s − 1.23e5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.894 + 0.447i)5-s + 0.795i·7-s − 0.333·9-s − 0.904·11-s + 0.382i·13-s + (0.258 − 0.516i)15-s + 1.01i·17-s + 0.846·19-s + 0.459·21-s + 1.14i·23-s + (0.599 + 0.799i)25-s + 0.192i·27-s + 1.16·29-s − 0.746·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(18.7431\)
Root analytic conductor: \(4.32933\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :7/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.55377 + 0.960286i\)
\(L(\frac12)\) \(\approx\) \(1.55377 + 0.960286i\)
\(L(\frac{9}{2})\) 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 \( 1 + 27iT \)
5 \( 1 + (-250 - 125i)T \)
good7 \( 1 - 722iT - 8.23e5T^{2} \)
11 \( 1 + 3.99e3T + 1.94e7T^{2} \)
13 \( 1 - 3.03e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.05e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.53e4T + 8.93e8T^{2} \)
23 \( 1 - 6.66e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.52e5T + 1.72e10T^{2} \)
31 \( 1 + 1.23e5T + 2.75e10T^{2} \)
37 \( 1 - 3.37e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.96e5T + 1.94e11T^{2} \)
43 \( 1 - 4.42e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.70e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.23e6iT - 1.17e12T^{2} \)
59 \( 1 - 3.02e5T + 2.48e12T^{2} \)
61 \( 1 + 2.83e6T + 3.14e12T^{2} \)
67 \( 1 + 3.74e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.00e6T + 9.09e12T^{2} \)
73 \( 1 - 2.40e6iT - 1.10e13T^{2} \)
79 \( 1 + 7.51e6T + 1.92e13T^{2} \)
83 \( 1 + 5.29e6iT - 2.71e13T^{2} \)
89 \( 1 - 7.65e6T + 4.42e13T^{2} \)
97 \( 1 - 1.00e7iT - 8.07e13T^{2} \)
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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

−13.69907881933056905380746212461, −12.83098248408564801401496230937, −11.62104191358790046833974538432, −10.33701289467929913579551276087, −9.135646224620786249404200720770, −7.76991128415772676410459069650, −6.34841656340094580033470517336, −5.33618482862279306802463284775, −2.92786830009403208169906627432, −1.64769298444566613509739289572, 0.68376097396052907161815049635, 2.72051907067689143569367878956, 4.57398884338298305398794221574, 5.68894304045933600996659773044, 7.38712905254115588818945735795, 8.884750044664893537439808047900, 10.02847936399806316095204157874, 10.78657078233898953820204984010, 12.41825149665973258205318738399, 13.57094730272544282526456505747

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