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Properties

Label 2-60-4.3-c4-0-13
Degree $2$
Conductor $60$
Sign $-0.606 + 0.795i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 − 3.78i)2-s + 5.19i·3-s + (−12.7 − 9.70i)4-s + 11.1·5-s + (19.6 + 6.65i)6-s − 89.0i·7-s + (−53.0 + 35.7i)8-s − 27·9-s + (14.3 − 42.3i)10-s − 174. i·11-s + (50.4 − 66.1i)12-s − 22.9·13-s + (−337. − 114. i)14-s + 58.0i·15-s + (67.7 + 246. i)16-s + 69.2·17-s + ⋯
L(s)  = 1  + (0.320 − 0.947i)2-s + 0.577i·3-s + (−0.795 − 0.606i)4-s + 0.447·5-s + (0.546 + 0.184i)6-s − 1.81i·7-s + (−0.828 + 0.559i)8-s − 0.333·9-s + (0.143 − 0.423i)10-s − 1.44i·11-s + (0.350 − 0.459i)12-s − 0.136·13-s + (−1.72 − 0.581i)14-s + 0.258i·15-s + (0.264 + 0.964i)16-s + 0.239·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.606 + 0.795i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.706712 - 1.42763i\)
\(L(\frac12)\) \(\approx\) \(0.706712 - 1.42763i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 3.78i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 - 11.1T \)
good7 \( 1 + 89.0iT - 2.40e3T^{2} \)
11 \( 1 + 174. iT - 1.46e4T^{2} \)
13 \( 1 + 22.9T + 2.85e4T^{2} \)
17 \( 1 - 69.2T + 8.35e4T^{2} \)
19 \( 1 - 341. iT - 1.30e5T^{2} \)
23 \( 1 + 319. iT - 2.79e5T^{2} \)
29 \( 1 - 679.T + 7.07e5T^{2} \)
31 \( 1 - 72.5iT - 9.23e5T^{2} \)
37 \( 1 - 2.37e3T + 1.87e6T^{2} \)
41 \( 1 + 762.T + 2.82e6T^{2} \)
43 \( 1 - 3.11e3iT - 3.41e6T^{2} \)
47 \( 1 - 315. iT - 4.87e6T^{2} \)
53 \( 1 - 3.38e3T + 7.89e6T^{2} \)
59 \( 1 + 6.68e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.31e3T + 1.38e7T^{2} \)
67 \( 1 - 4.01e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.95e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.74e3T + 2.83e7T^{2} \)
79 \( 1 - 414. iT - 3.89e7T^{2} \)
83 \( 1 + 9.73e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.19e3T + 6.27e7T^{2} \)
97 \( 1 - 9.56e3T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.92823728469327804534215537987, −12.97396042938898750129031188368, −11.35089708567955918911802540642, −10.52904327329882773285615449814, −9.756976767075720798335948651591, −8.209371976110674248132737873545, −6.11819664361909012811233845213, −4.49392683477974391418041522384, −3.29253593446324738174458430149, −0.861185927688383303501649275142, 2.46951556591074284626432828682, 4.97615569060693608408630531292, 6.06656156822570787632575270940, 7.30595950126538272850208281775, 8.699262906384573374810425490572, 9.612487966069853349937450841542, 11.93018347063258401715874339314, 12.57033871560986145687685749092, 13.66259963654425369780771995759, 15.02488758791356868064401176858

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