↑

Properties

Label 2-60-15.2-c5-0-8
Degree $2$
Conductor $60$
Sign $0.256 + 0.966i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (14.8 − 4.61i)3-s + (31.2 − 46.3i)5-s + (−91.3 − 91.3i)7-s + (200. − 137. i)9-s − 99.2i·11-s + (−452. + 452. i)13-s + (252. − 834. i)15-s + (1.26e3 − 1.26e3i)17-s − 912. i·19-s + (−1.78e3 − 939. i)21-s + (1.74e3 + 1.74e3i)23-s + (−1.16e3 − 2.89e3i)25-s + (2.35e3 − 2.97e3i)27-s + 299.·29-s + 2.59e3·31-s + ⋯
L(s)  = 1  + (0.955 − 0.295i)3-s + (0.559 − 0.828i)5-s + (−0.705 − 0.705i)7-s + (0.824 − 0.565i)9-s − 0.247i·11-s + (−0.742 + 0.742i)13-s + (0.289 − 0.957i)15-s + (1.06 − 1.06i)17-s − 0.580i·19-s + (−0.882 − 0.464i)21-s + (0.688 + 0.688i)23-s + (−0.373 − 0.927i)25-s + (0.620 − 0.784i)27-s + 0.0661·29-s + 0.484·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.83777 - 1.41315i\)
\(L(\frac12)\) \(\approx\) \(1.83777 - 1.41315i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-14.8 + 4.61i)T \)
5 \( 1 + (-31.2 + 46.3i)T \)
good7 \( 1 + (91.3 + 91.3i)T + 1.68e4iT^{2} \)
11 \( 1 + 99.2iT - 1.61e5T^{2} \)
13 \( 1 + (452. - 452. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.26e3 + 1.26e3i)T - 1.41e6iT^{2} \)
19 \( 1 + 912. iT - 2.47e6T^{2} \)
23 \( 1 + (-1.74e3 - 1.74e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 299.T + 2.05e7T^{2} \)
31 \( 1 - 2.59e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.91e3 - 5.91e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.94e4iT - 1.15e8T^{2} \)
43 \( 1 + (8.60e3 - 8.60e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.73e4 - 1.73e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.94e4 - 1.94e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 220.T + 7.14e8T^{2} \)
61 \( 1 - 4.39e4T + 8.44e8T^{2} \)
67 \( 1 + (3.86e4 + 3.86e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.18e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.40e4 - 1.40e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 6.48e4iT - 3.07e9T^{2} \)
83 \( 1 + (-5.90e3 - 5.90e3i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + (7.06e4 + 7.06e4i)T + 8.58e9iT^{2} \)
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

−13.65868166645406787512290490014, −13.09829226148207187004486358105, −11.84192559448442714097256113278, −9.830291479889281815858731539652, −9.346135532997099164335483463158, −7.86676324822937589361303426642, −6.64413803350933444984712726506, −4.71471717803099341975768934110, −2.96965198017690813920923352732, −1.07941628758719541280200705579, 2.29388131653503489712154546357, 3.46418589006882639448598991381, 5.59558342614273312284451785246, 7.11086111050712711046538308421, 8.484439688846264006400908360581, 9.834756940432595583556395057476, 10.38740374768155903701331698780, 12.33450936978251578031447033068, 13.27347313075061303431197007143, 14.67808629969956234261603389158

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