↑

Properties

Label 2-60-60.23-c2-0-2
Degree $2$
Conductor $60$
Sign $0.741 - 0.671i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 + 0.770i)2-s + (−2.78 − 1.12i)3-s + (2.81 − 2.84i)4-s + (3.86 + 3.17i)5-s + (5.99 − 0.0674i)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 + 4.74i)12-s + (−4.22 − 4.22i)13-s + (−12.4 − 5.10i)14-s + (−7.16 − 13.1i)15-s + (−0.188 − 15.9i)16-s + (9.35 + 9.35i)17-s + ⋯
L(s)  = 1  + (−0.922 + 0.385i)2-s + (−0.927 − 0.375i)3-s + (0.702 − 0.711i)4-s + (0.772 + 0.635i)5-s + (0.999 − 0.0112i)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 + (−0.918 + 0.395i)12-s + (−0.324 − 0.324i)13-s + (−0.887 − 0.364i)14-s + (−0.477 − 0.878i)15-s + (−0.0117 − 0.999i)16-s + (0.550 + 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.741 - 0.671i$
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: \((2,\ 60,\ (\ :1),\ 0.741 - 0.671i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.694717 + 0.267871i\)
\(L(\frac12)\) \(\approx\) \(0.694717 + 0.267871i\)
\(L(2)\) 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 \( 1 + (2.78 + 1.12i)T \)
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 \( 1 + (-6.90 + 6.90i)T - 9.40e3iT^{2} \)
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

−15.00112980514837189452218952493, −14.22481896077334585843445012478, −12.41521075742851717161634919307, −11.32871115227362518385076357199, −10.41026492020859946672006613814, −9.188714460105212338432652189082, −7.62680158766522626404470769083, −6.38468017967639571489463213234, −5.46612867276007839140388153358, −1.80968155185674960950717824300, 1.28227796038914560325935160389, 4.31468332075551157875556238753, 6.08178404748233135287049599362, 7.55499385392636211904273534664, 9.284640083347564952955609913558, 9.948323585055286245297239179299, 11.29008096294566333213791522389, 11.99084305344021805001338293800, 13.36565748726733673155587336878, 14.89560130831574644853677724593

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