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Properties

Label 2-60-15.2-c3-0-0
Degree $2$
Conductor $60$
Sign $-0.886 - 0.463i$
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  + (−1.18 + 5.05i)3-s + (−8.06 + 7.74i)5-s + (−16.2 − 16.2i)7-s + (−24.1 − 11.9i)9-s + 70.9i·11-s + (2.51 − 2.51i)13-s + (−29.6 − 49.9i)15-s + (24.8 − 24.8i)17-s + 114. i·19-s + (101. − 62.9i)21-s + (45.7 + 45.7i)23-s + (4.99 − 124. i)25-s + (89.3 − 108. i)27-s − 42.0·29-s + 193.·31-s + ⋯
L(s)  = 1  + (−0.228 + 0.973i)3-s + (−0.721 + 0.692i)5-s + (−0.877 − 0.877i)7-s + (−0.895 − 0.444i)9-s + 1.94i·11-s + (0.0535 − 0.0535i)13-s + (−0.509 − 0.860i)15-s + (0.354 − 0.354i)17-s + 1.38i·19-s + (1.05 − 0.653i)21-s + (0.414 + 0.414i)23-s + (0.0399 − 0.999i)25-s + (0.637 − 0.770i)27-s − 0.269·29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.886 - 0.463i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.886 - 0.463i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.170471 + 0.693443i\)
\(L(\frac12)\) \(\approx\) \(0.170471 + 0.693443i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.18 - 5.05i)T \)
5 \( 1 + (8.06 - 7.74i)T \)
good7 \( 1 + (16.2 + 16.2i)T + 343iT^{2} \)
11 \( 1 - 70.9iT - 1.33e3T^{2} \)
13 \( 1 + (-2.51 + 2.51i)T - 2.19e3iT^{2} \)
17 \( 1 + (-24.8 + 24.8i)T - 4.91e3iT^{2} \)
19 \( 1 - 114. iT - 6.85e3T^{2} \)
23 \( 1 + (-45.7 - 45.7i)T + 1.21e4iT^{2} \)
29 \( 1 + 42.0T + 2.43e4T^{2} \)
31 \( 1 - 193.T + 2.97e4T^{2} \)
37 \( 1 + (37.4 + 37.4i)T + 5.06e4iT^{2} \)
41 \( 1 - 245. iT - 6.89e4T^{2} \)
43 \( 1 + (171. - 171. i)T - 7.95e4iT^{2} \)
47 \( 1 + (253. - 253. i)T - 1.03e5iT^{2} \)
53 \( 1 + (224. + 224. i)T + 1.48e5iT^{2} \)
59 \( 1 + 225.T + 2.05e5T^{2} \)
61 \( 1 - 183.T + 2.26e5T^{2} \)
67 \( 1 + (-316. - 316. i)T + 3.00e5iT^{2} \)
71 \( 1 + 225. iT - 3.57e5T^{2} \)
73 \( 1 + (-349. + 349. i)T - 3.89e5iT^{2} \)
79 \( 1 - 323. iT - 4.93e5T^{2} \)
83 \( 1 + (553. + 553. i)T + 5.71e5iT^{2} \)
89 \( 1 - 351.T + 7.04e5T^{2} \)
97 \( 1 + (114. + 114. i)T + 9.12e5iT^{2} \)
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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

−15.13048021917023741043739972710, −14.29879823986700939617743920967, −12.65549678919212013025471271213, −11.54766863087831300133501824229, −10.19396214367276682740976158360, −9.777910322434888076808581303889, −7.74668224695670117284896396432, −6.55309851262963085632876038698, −4.56070074400672657363889042776, −3.38300840617126690506189526069, 0.51133657051135851339473983402, 3.08980769644256987671748194607, 5.44830576448331229225507919344, 6.61363932073013039823529234640, 8.256981965458570328746168299942, 8.962914565930602607935645741410, 11.08061159036174983662017109428, 11.97136239071791332549699651386, 12.93132546498578556559804426160, 13.76111341659799676149274716548

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