Properties

Label 2-60-4.3-c2-0-4
Degree $2$
Conductor $60$
Sign $-0.168 + 0.985i$
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  + (−1.99 − 0.169i)2-s − 1.73i·3-s + (3.94 + 0.675i)4-s − 2.23·5-s + (−0.293 + 3.45i)6-s − 12.3i·7-s + (−7.74 − 2.01i)8-s − 2.99·9-s + (4.45 + 0.378i)10-s − 11.0i·11-s + (1.16 − 6.82i)12-s + 2.82·13-s + (−2.10 + 24.7i)14-s + 3.87i·15-s + (15.0 + 5.32i)16-s + 6.52·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0847i)2-s − 0.577i·3-s + (0.985 + 0.168i)4-s − 0.447·5-s + (−0.0489 + 0.575i)6-s − 1.77i·7-s + (−0.967 − 0.251i)8-s − 0.333·9-s + (0.445 + 0.0378i)10-s − 1.00i·11-s + (0.0974 − 0.569i)12-s + 0.216·13-s + (−0.150 + 1.76i)14-s + 0.258i·15-s + (0.942 + 0.332i)16-s + 0.383·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.168 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.439081 - 0.520690i\)
\(L(\frac12)\) \(\approx\) \(0.439081 - 0.520690i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 0.169i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 + 2.23T \)
good7 \( 1 + 12.3iT - 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 - 2.82T + 169T^{2} \)
17 \( 1 - 6.52T + 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 - 7.90iT - 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 + 36.3iT - 961T^{2} \)
37 \( 1 + 18.9T + 1.36e3T^{2} \)
41 \( 1 - 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 + 11.7iT - 2.20e3T^{2} \)
53 \( 1 - 41.1T + 2.80e3T^{2} \)
59 \( 1 + 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 + 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6T + 5.32e3T^{2} \)
79 \( 1 + 123. iT - 6.24e3T^{2} \)
83 \( 1 - 99.6iT - 6.88e3T^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−14.47070702369590814677068954502, −13.42713369711482400220793139329, −12.01724500389674446734955490783, −10.94686069517037507659369712518, −10.04129373878097125496632916885, −8.291676221225078395612660774443, −7.55242784549543061942805492855, −6.30976322595577054641652051764, −3.58494888069842072116435183677, −0.930938434974252763118980201957, 2.64985865776657611086381695048, 5.17016178522101669360572178248, 6.76825884981132550636233536522, 8.432746224723771176568181481205, 9.157379357223260356592991704964, 10.37393153823543199584035584427, 11.69025814133452859123467926836, 12.39221254608749833453169980403, 14.62350742297007000648936191790, 15.57287476561665567986187532731

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