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Properties

Label 2-60-5.2-c2-0-0
Degree $2$
Conductor $60$
Sign $0.714 - 0.699i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + (4.22 + 2.67i)5-s + (7.44 + 7.44i)7-s − 2.99i·9-s − 16.2·11-s + (12.2 − 12.2i)13-s + (−8.44 + 1.89i)15-s + (−7.55 − 7.55i)17-s − 14.4i·19-s − 18.2·21-s + (−2.65 + 2.65i)23-s + (10.6 + 22.5i)25-s + (3.67 + 3.67i)27-s − 34.2i·29-s − 20.4·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.844 + 0.534i)5-s + (1.06 + 1.06i)7-s − 0.333i·9-s − 1.47·11-s + (0.942 − 0.942i)13-s + (−0.563 + 0.126i)15-s + (−0.444 − 0.444i)17-s − 0.762i·19-s − 0.868·21-s + (−0.115 + 0.115i)23-s + (0.427 + 0.903i)25-s + (0.136 + 0.136i)27-s − 1.18i·29-s − 0.661·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.714 - 0.699i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.12330 + 0.458226i\)
\(L(\frac12)\) \(\approx\) \(1.12330 + 0.458226i\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (-4.22 - 2.67i)T \)
good7 \( 1 + (-7.44 - 7.44i)T + 49iT^{2} \)
11 \( 1 + 16.2T + 121T^{2} \)
13 \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \)
17 \( 1 + (7.55 + 7.55i)T + 289iT^{2} \)
19 \( 1 + 14.4iT - 361T^{2} \)
23 \( 1 + (2.65 - 2.65i)T - 529iT^{2} \)
29 \( 1 + 34.2iT - 841T^{2} \)
31 \( 1 + 20.4T + 961T^{2} \)
37 \( 1 + (-7.34 - 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 - 25.5T + 1.68e3T^{2} \)
43 \( 1 + (-25.1 + 25.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-22.0 - 22.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (35.3 - 35.3i)T - 2.80e3iT^{2} \)
59 \( 1 + 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (24.6 + 24.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 77.9T + 5.04e3T^{2} \)
73 \( 1 + (44.1 - 44.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 48.4iT - 6.24e3T^{2} \)
83 \( 1 + (101. - 101. i)T - 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (-55.4 - 55.4i)T + 9.40e3iT^{2} \)
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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

−15.26987712515171750160724499769, −13.88876682523468218955548846378, −12.81088761327700831919043939412, −11.26360549456479822717955153529, −10.61940021839009609489343185320, −9.221575252487642135957640942607, −7.88530539513402913080314698975, −5.95568981056281887558305060510, −5.09713493725850525374420201348, −2.56332962759671922337603389589, 1.61327538530376155463047811506, 4.56025337077923771172452644817, 5.88571435126036403652192656293, 7.46576605683947442746350224208, 8.662762912871997279109339698853, 10.38333772758251944639391771482, 11.10786837151392328385220998428, 12.68834342888117124959265272119, 13.55164573360286998730456637628, 14.37147059380214155716697863083

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