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Properties

Label 2-60-5.2-c4-0-3
Degree $2$
Conductor $60$
Sign $-0.694 + 0.719i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.67 − 3.67i)3-s + (−21.4 − 12.7i)5-s + (−29.2 − 29.2i)7-s − 27i·9-s − 100.·11-s + (65.5 − 65.5i)13-s + (−125. + 32.0i)15-s + (−206. − 206. i)17-s − 260. i·19-s − 214.·21-s + (210. − 210. i)23-s + (299. + 548. i)25-s + (−99.2 − 99.2i)27-s + 503. i·29-s + 1.25e3·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.859 − 0.510i)5-s + (−0.596 − 0.596i)7-s − 0.333i·9-s − 0.833·11-s + (0.387 − 0.387i)13-s + (−0.559 + 0.142i)15-s + (−0.713 − 0.713i)17-s − 0.720i·19-s − 0.487·21-s + (0.397 − 0.397i)23-s + (0.478 + 0.878i)25-s + (−0.136 − 0.136i)27-s + 0.598i·29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.694 + 0.719i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.391023 - 0.920828i\)
\(L(\frac12)\) \(\approx\) \(0.391023 - 0.920828i\)
\(L(3)\) 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 \( 1 + (-3.67 + 3.67i)T \)
5 \( 1 + (21.4 + 12.7i)T \)
good7 \( 1 + (29.2 + 29.2i)T + 2.40e3iT^{2} \)
11 \( 1 + 100.T + 1.46e4T^{2} \)
13 \( 1 + (-65.5 + 65.5i)T - 2.85e4iT^{2} \)
17 \( 1 + (206. + 206. i)T + 8.35e4iT^{2} \)
19 \( 1 + 260. iT - 1.30e5T^{2} \)
23 \( 1 + (-210. + 210. i)T - 2.79e5iT^{2} \)
29 \( 1 - 503. iT - 7.07e5T^{2} \)
31 \( 1 - 1.25e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.81e3 - 1.81e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 3.12e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.14e3 + 2.14e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-838. - 838. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-883. + 883. i)T - 7.89e6iT^{2} \)
59 \( 1 + 4.93e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.87e3T + 1.38e7T^{2} \)
67 \( 1 + (932. + 932. i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.43e3T + 2.54e7T^{2} \)
73 \( 1 + (412. - 412. i)T - 2.83e7iT^{2} \)
79 \( 1 + 8.65e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.22e3 + 5.22e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.10e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.41e3 + 2.41e3i)T + 8.85e7iT^{2} \)
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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

−13.58655168756908980969754858683, −13.01542210792933078601595779612, −11.77227424286022519672372444493, −10.48883306714617878730865445395, −8.984751188964449641277373021338, −7.88568436031409231674103396347, −6.75462391632310267745815245890, −4.72528322417607629606663158475, −3.07404111189400603123428895567, −0.53093047810948250745902207118, 2.77130912119517751410026457508, 4.20114517711439087711634581677, 6.12143641299455294372560664182, 7.69616788151646361443527766937, 8.811402247206614551955101432852, 10.17751689938961497996478576876, 11.25865966315863690895815183824, 12.50341516665388569615222275460, 13.69291071461434125671645750761, 15.07069945654756307874309898755

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