↑

Properties

Label 2-60-12.11-c3-0-22
Degree $2$
Conductor $60$
Sign $-0.325 + 0.945i$
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  + (1.91 − 2.08i)2-s + (2.09 − 4.75i)3-s + (−0.661 − 7.97i)4-s − 5i·5-s + (−5.88 − 13.4i)6-s + 32.2i·7-s + (−17.8 − 13.8i)8-s + (−18.2 − 19.9i)9-s + (−10.4 − 9.57i)10-s + 32.4·11-s + (−39.3 − 13.5i)12-s + 44.9·13-s + (67.1 + 61.8i)14-s + (−23.7 − 10.4i)15-s + (−63.1 + 10.5i)16-s − 79.2i·17-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)2-s + (0.402 − 0.915i)3-s + (−0.0826 − 0.996i)4-s − 0.447i·5-s + (−0.400 − 0.916i)6-s + 1.74i·7-s + (−0.789 − 0.614i)8-s + (−0.675 − 0.737i)9-s + (−0.329 − 0.302i)10-s + 0.890·11-s + (−0.945 − 0.325i)12-s + 0.958·13-s + (1.28 + 1.18i)14-s + (−0.409 − 0.180i)15-s + (−0.986 + 0.164i)16-s − 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.325 + 0.945i$
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: \((2,\ 60,\ (\ :3/2),\ -0.325 + 0.945i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.24989 - 1.75247i\)
\(L(\frac12)\) \(\approx\) \(1.24989 - 1.75247i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.91 + 2.08i)T \)
3 \( 1 + (-2.09 + 4.75i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 32.2iT - 343T^{2} \)
11 \( 1 - 32.4T + 1.33e3T^{2} \)
13 \( 1 - 44.9T + 2.19e3T^{2} \)
17 \( 1 + 79.2iT - 4.91e3T^{2} \)
19 \( 1 - 69.0iT - 6.85e3T^{2} \)
23 \( 1 - 72.8T + 1.21e4T^{2} \)
29 \( 1 - 219. iT - 2.43e4T^{2} \)
31 \( 1 - 28.4iT - 2.97e4T^{2} \)
37 \( 1 + 152.T + 5.06e4T^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 - 96.6iT - 7.95e4T^{2} \)
47 \( 1 + 103.T + 1.03e5T^{2} \)
53 \( 1 + 544. iT - 1.48e5T^{2} \)
59 \( 1 + 431.T + 2.05e5T^{2} \)
61 \( 1 - 183.T + 2.26e5T^{2} \)
67 \( 1 + 290. iT - 3.00e5T^{2} \)
71 \( 1 + 489.T + 3.57e5T^{2} \)
73 \( 1 + 292.T + 3.89e5T^{2} \)
79 \( 1 - 656. iT - 4.93e5T^{2} \)
83 \( 1 - 392.T + 5.71e5T^{2} \)
89 \( 1 - 336. iT - 7.04e5T^{2} \)
97 \( 1 - 1.03e3T + 9.12e5T^{2} \)
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

−14.09038573652599330066258261147, −12.91431231463147612823445452141, −12.14235842698686541360408833894, −11.41860590198228226699727708070, −9.323274040511496005420700109711, −8.637909890177635246807336384705, −6.50647917321245723840222728474, −5.36140133758919726993101089971, −3.17555005072453670313342068306, −1.57775646734843502769992366811, 3.56001043001767997817411821930, 4.36013804432138581908631979008, 6.29113362834333998220074463307, 7.56349818836903059307548306799, 8.886833309488108812320675001404, 10.42493782367679281742929065407, 11.36659723337015936917384526629, 13.31148341478588603964825308228, 13.93656105905239687855403357434, 14.87555017924609265330391285528

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