↑

Properties

Label 2-60-12.11-c3-0-19
Degree $2$
Conductor $60$
Sign $-0.991 + 0.132i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 0.245i)2-s + (−1.56 − 4.95i)3-s + (7.87 + 1.38i)4-s − 5i·5-s + (3.19 + 14.3i)6-s − 5.80i·7-s + (−21.8 − 5.83i)8-s + (−22.0 + 15.5i)9-s + (−1.22 + 14.0i)10-s − 27.6·11-s + (−5.49 − 41.2i)12-s − 70.9·13-s + (−1.42 + 16.3i)14-s + (−24.7 + 7.83i)15-s + (60.1 + 21.8i)16-s − 89.8i·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0868i)2-s + (−0.301 − 0.953i)3-s + (0.984 + 0.172i)4-s − 0.447i·5-s + (0.217 + 0.976i)6-s − 0.313i·7-s + (−0.966 − 0.257i)8-s + (−0.818 + 0.574i)9-s + (−0.0388 + 0.445i)10-s − 0.757·11-s + (−0.132 − 0.991i)12-s − 1.51·13-s + (−0.0272 + 0.312i)14-s + (−0.426 + 0.134i)15-s + (0.940 + 0.340i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.991 + 0.132i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.991 + 0.132i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0285605 - 0.430611i\)
\(L(\frac12)\) \(\approx\) \(0.0285605 - 0.430611i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 + 0.245i)T \)
3 \( 1 + (1.56 + 4.95i)T \)
5 \( 1 + 5iT \)
good7 \( 1 + 5.80iT - 343T^{2} \)
11 \( 1 + 27.6T + 1.33e3T^{2} \)
13 \( 1 + 70.9T + 2.19e3T^{2} \)
17 \( 1 + 89.8iT - 4.91e3T^{2} \)
19 \( 1 - 68.6iT - 6.85e3T^{2} \)
23 \( 1 - 73.5T + 1.21e4T^{2} \)
29 \( 1 - 18.2iT - 2.43e4T^{2} \)
31 \( 1 + 277. iT - 2.97e4T^{2} \)
37 \( 1 + 26.7T + 5.06e4T^{2} \)
41 \( 1 + 364. iT - 6.89e4T^{2} \)
43 \( 1 + 425. iT - 7.95e4T^{2} \)
47 \( 1 + 80.9T + 1.03e5T^{2} \)
53 \( 1 - 610. iT - 1.48e5T^{2} \)
59 \( 1 + 522.T + 2.05e5T^{2} \)
61 \( 1 - 372.T + 2.26e5T^{2} \)
67 \( 1 + 369. iT - 3.00e5T^{2} \)
71 \( 1 - 985.T + 3.57e5T^{2} \)
73 \( 1 + 2.29T + 3.89e5T^{2} \)
79 \( 1 + 175. iT - 4.93e5T^{2} \)
83 \( 1 - 207.T + 5.71e5T^{2} \)
89 \( 1 + 186. iT - 7.04e5T^{2} \)
97 \( 1 + 1.24e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.91858754257233614110996491136, −12.58611901908173343839685908753, −11.83873205479176972594605089285, −10.55318813967871902576536290127, −9.286648283050627958202870325013, −7.82816798788929741557322292072, −7.13629036140465947863510349623, −5.42611079886178423133080137045, −2.39038381611008090177693330340, −0.40073450769488810159510399390, 2.82291206435754408696075329380, 5.15375613237905335012847678991, 6.68432582933024416987750572967, 8.202583312790892802937282855459, 9.497535973251769247340181543422, 10.38369476737688172345312205986, 11.27193009999710157473948000532, 12.52493412417763303915671291534, 14.67026382604623146655059429096, 15.21983540526696735195540601352

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