↑

Properties

Label 2-60-60.59-c3-0-23
Degree $2$
Conductor $60$
Sign $0.773 + 0.633i$
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  + (2.76 − 0.583i)2-s + (3.16 − 4.12i)3-s + (7.31 − 3.22i)4-s + (−4.82 + 10.0i)5-s + (6.35 − 13.2i)6-s + 10.1·7-s + (18.3 − 13.2i)8-s + (−6.95 − 26.0i)9-s + (−7.47 + 30.7i)10-s − 60.6·11-s + (9.86 − 40.3i)12-s + 26.6i·13-s + (28.0 − 5.91i)14-s + (26.2 + 51.8i)15-s + (43.1 − 47.2i)16-s + 32.4·17-s + ⋯
L(s)  = 1  + (0.978 − 0.206i)2-s + (0.609 − 0.793i)3-s + (0.914 − 0.403i)4-s + (−0.431 + 0.901i)5-s + (0.432 − 0.901i)6-s + 0.547·7-s + (0.812 − 0.583i)8-s + (−0.257 − 0.966i)9-s + (−0.236 + 0.971i)10-s − 1.66·11-s + (0.237 − 0.971i)12-s + 0.567i·13-s + (0.536 − 0.112i)14-s + (0.452 + 0.891i)15-s + (0.674 − 0.738i)16-s + 0.463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.52788 - 0.902764i\)
\(L(\frac12)\) \(\approx\) \(2.52788 - 0.902764i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.76 + 0.583i)T \)
3 \( 1 + (-3.16 + 4.12i)T \)
5 \( 1 + (4.82 - 10.0i)T \)
good7 \( 1 - 10.1T + 343T^{2} \)
11 \( 1 + 60.6T + 1.33e3T^{2} \)
13 \( 1 - 26.6iT - 2.19e3T^{2} \)
17 \( 1 - 32.4T + 4.91e3T^{2} \)
19 \( 1 - 106. iT - 6.85e3T^{2} \)
23 \( 1 - 88.0iT - 1.21e4T^{2} \)
29 \( 1 - 101. iT - 2.43e4T^{2} \)
31 \( 1 + 103. iT - 2.97e4T^{2} \)
37 \( 1 + 331. iT - 5.06e4T^{2} \)
41 \( 1 - 13.7iT - 6.89e4T^{2} \)
43 \( 1 - 72.5T + 7.95e4T^{2} \)
47 \( 1 + 463. iT - 1.03e5T^{2} \)
53 \( 1 - 282.T + 1.48e5T^{2} \)
59 \( 1 - 682.T + 2.05e5T^{2} \)
61 \( 1 - 140.T + 2.26e5T^{2} \)
67 \( 1 + 515.T + 3.00e5T^{2} \)
71 \( 1 + 38.7T + 3.57e5T^{2} \)
73 \( 1 + 747. iT - 3.89e5T^{2} \)
79 \( 1 - 862. iT - 4.93e5T^{2} \)
83 \( 1 - 534. iT - 5.71e5T^{2} \)
89 \( 1 + 507. iT - 7.04e5T^{2} \)
97 \( 1 - 376. iT - 9.12e5T^{2} \)
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

−14.38535930642341192727585237742, −13.46549722983279020586814975225, −12.36348628040290646192496714213, −11.34495541375759712196189021346, −10.18249085100477043474356153929, −7.974859438468691734474796204009, −7.19749632563795048551716832467, −5.65673296868987034423314089896, −3.61939650673323382438208185893, −2.20026650895739527110258191369, 2.81552975650201617027698850085, 4.56256650651181013954073706934, 5.30553307745252649652602867667, 7.69835389273183704485149372582, 8.468856138152796277497369283934, 10.28839363236866223629218860857, 11.38944296793882020856692028263, 12.81210770874887159769372472938, 13.55769269368333147423755420476, 14.87072353491953282327387115979

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