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Properties

Label 2-60-20.19-c6-0-15
Degree $2$
Conductor $60$
Sign $-0.506 - 0.862i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.294 + 7.99i)2-s + 15.5·3-s + (−63.8 − 4.71i)4-s + (71.1 + 102. i)5-s + (−4.59 + 124. i)6-s + 496.·7-s + (56.5 − 508. i)8-s + 243·9-s + (−842. + 538. i)10-s + 561. i·11-s + (−994. − 73.5i)12-s + 1.58e3i·13-s + (−146. + 3.96e3i)14-s + (1.10e3 + 1.60e3i)15-s + (4.05e3 + 601. i)16-s − 4.50e3i·17-s + ⋯
L(s)  = 1  + (−0.0368 + 0.999i)2-s + 0.577·3-s + (−0.997 − 0.0736i)4-s + (0.569 + 0.822i)5-s + (−0.0212 + 0.576i)6-s + 1.44·7-s + (0.110 − 0.993i)8-s + 0.333·9-s + (−0.842 + 0.538i)10-s + 0.421i·11-s + (−0.575 − 0.0425i)12-s + 0.721i·13-s + (−0.0533 + 1.44i)14-s + (0.328 + 0.474i)15-s + (0.989 + 0.146i)16-s − 0.916i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.17369 + 2.05168i\)
\(L(\frac12)\) \(\approx\) \(1.17369 + 2.05168i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.294 - 7.99i)T \)
3 \( 1 - 15.5T \)
5 \( 1 + (-71.1 - 102. i)T \)
good7 \( 1 - 496.T + 1.17e5T^{2} \)
11 \( 1 - 561. iT - 1.77e6T^{2} \)
13 \( 1 - 1.58e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.50e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.01e4iT - 4.70e7T^{2} \)
23 \( 1 + 1.61e4T + 1.48e8T^{2} \)
29 \( 1 - 2.00e3T + 5.94e8T^{2} \)
31 \( 1 - 7.38e3iT - 8.87e8T^{2} \)
37 \( 1 + 6.66e4iT - 2.56e9T^{2} \)
41 \( 1 + 9.71e4T + 4.75e9T^{2} \)
43 \( 1 - 1.12e5T + 6.32e9T^{2} \)
47 \( 1 - 1.96e5T + 1.07e10T^{2} \)
53 \( 1 + 7.20e4iT - 2.21e10T^{2} \)
59 \( 1 - 1.41e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.01e5T + 5.15e10T^{2} \)
67 \( 1 - 2.16e5T + 9.04e10T^{2} \)
71 \( 1 + 3.58e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.11e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.72e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.52e5T + 3.26e11T^{2} \)
89 \( 1 + 8.72e4T + 4.96e11T^{2} \)
97 \( 1 - 1.30e5iT - 8.32e11T^{2} \)
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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

−14.19499505462981733528170394681, −13.90294718913921152705785420541, −12.08842306919051552855086475586, −10.47893968801530279537895540898, −9.341163015812917720225252999126, −8.041327397363663459842192883975, −7.12629142290136269428856598785, −5.61718730642207482638788399115, −4.11479750091565754632974395599, −1.88202370985366428029155709075, 1.05067235725082536188564168463, 2.31052733186608465219307217810, 4.24606078712449897836959434784, 5.42716147532601393893037433909, 8.101533912761670924424029681994, 8.706970113175909860167193501831, 10.03556272045975489449725306952, 11.16833505276637483872576514907, 12.36848648281031523917768420943, 13.45318451977328838780850406257

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