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Properties

Label 2-60-60.59-c3-0-7
Degree $2$
Conductor $60$
Sign $-0.0537 - 0.998i$
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

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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 + (−11.1 + 9.55i)6-s − 10.1·7-s + (18.3 + 13.2i)8-s + (−6.95 − 26.0i)9-s + (−19.2 + 25.0i)10-s + 60.6·11-s + (−36.4 + 19.9i)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 + (−0.759 + 0.650i)6-s − 0.547·7-s + (0.812 + 0.583i)8-s + (−0.257 − 0.966i)9-s + (−0.608 + 0.793i)10-s + 1.66·11-s + (−0.877 + 0.479i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0537 - 0.998i$
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: \((2,\ 60,\ (\ :3/2),\ -0.0537 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33574 + 1.40963i\)
\(L(\frac12)\) \(\approx\) \(1.33574 + 1.40963i\)
\(L(\frac{5}{2})\) 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 \( 1 - 376. iT - 9.12e5T^{2} \)
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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

−14.76398328266066270661251209123, −14.17104032815673983287810663257, −12.42601737737747221598912070181, −11.54585955045029275428495843996, −10.70132166932387052044361894596, −9.187457070089776071947005526265, −6.97216954264341978685655980376, −6.24176500685139353743976972638, −4.44517022668398700891363500293, −3.31296523830883521095863291333, 1.30034224118958620444866150872, 3.82321042173119837795148653477, 5.46554572654685698484992447267, 6.56682693372259918730270558696, 7.975935079206575176966798214373, 9.865532354522956276023384012833, 11.56166187794346523397928669073, 12.10457707428422678298746613961, 12.99525516927809710798726008417, 13.98303888399603039129468132528

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