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Properties

Label 2-60-60.23-c2-0-10
Degree $2$
Conductor $60$
Sign $0.997 + 0.0659i$
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  + (1.84 − 0.770i)2-s + (1.12 + 2.78i)3-s + (2.81 − 2.84i)4-s + (−3.86 − 3.17i)5-s + (4.21 + 4.26i)6-s + (4.75 + 4.75i)7-s + (2.99 − 7.41i)8-s + (−6.46 + 6.25i)9-s + (−9.57 − 2.88i)10-s − 11.9·11-s + (11.0 + 4.61i)12-s + (−4.22 − 4.22i)13-s + (12.4 + 5.10i)14-s + (4.48 − 14.3i)15-s + (−0.188 − 15.9i)16-s + (−9.35 − 9.35i)17-s + ⋯
L(s)  = 1  + (0.922 − 0.385i)2-s + (0.375 + 0.927i)3-s + (0.702 − 0.711i)4-s + (−0.772 − 0.635i)5-s + (0.703 + 0.710i)6-s + (0.678 + 0.678i)7-s + (0.374 − 0.927i)8-s + (−0.718 + 0.695i)9-s + (−0.957 − 0.288i)10-s − 1.08·11-s + (0.922 + 0.384i)12-s + (−0.324 − 0.324i)13-s + (0.887 + 0.364i)14-s + (0.299 − 0.954i)15-s + (−0.0117 − 0.999i)16-s + (−0.550 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.997 + 0.0659i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.997 + 0.0659i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90118 - 0.0627229i\)
\(L(\frac12)\) \(\approx\) \(1.90118 - 0.0627229i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.84 + 0.770i)T \)
3 \( 1 + (-1.12 - 2.78i)T \)
5 \( 1 + (3.86 + 3.17i)T \)
good7 \( 1 + (-4.75 - 4.75i)T + 49iT^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 + (4.22 + 4.22i)T + 169iT^{2} \)
17 \( 1 + (9.35 + 9.35i)T + 289iT^{2} \)
19 \( 1 - 1.48T + 361T^{2} \)
23 \( 1 + (-11.6 - 11.6i)T + 529iT^{2} \)
29 \( 1 - 39.3T + 841T^{2} \)
31 \( 1 - 43.6iT - 961T^{2} \)
37 \( 1 + (-49.1 + 49.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 27.0iT - 1.68e3T^{2} \)
43 \( 1 + (-8.84 + 8.84i)T - 1.84e3iT^{2} \)
47 \( 1 + (-15.0 + 15.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 61.7iT - 3.48e3T^{2} \)
61 \( 1 + 84.6T + 3.72e3T^{2} \)
67 \( 1 + (65.7 + 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 14.2T + 5.04e3T^{2} \)
73 \( 1 + (-16.0 - 16.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 9.32T + 6.24e3T^{2} \)
83 \( 1 + (-12.7 - 12.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.4T + 7.92e3T^{2} \)
97 \( 1 + (-6.90 + 6.90i)T - 9.40e3iT^{2} \)
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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

−15.00139448571707837851327175907, −13.78588426919945023373571755267, −12.55500884146868796624156240103, −11.48917554068505891448407356111, −10.53591304729881810230543931371, −9.042760019582273908256683209233, −7.74166646239308158487263157424, −5.33665467345467504625567063008, −4.58322399178332770272575374183, −2.84064430313499975460764774856, 2.70772680971985933622496677122, 4.44924124240409045374123279407, 6.41358845222503665715798397139, 7.53794378892968101715187511744, 8.192786402010872288378790328877, 10.76365850755716623148203557688, 11.70385447650261247709424468715, 12.84845952260320675787205869778, 13.80841938450407407368815934379, 14.70472825882414882883551676309

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