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Properties

Label 2-60-15.8-c3-0-2
Degree $2$
Conductor $60$
Sign $0.999 + 0.0184i$
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  + (5.05 + 1.18i)3-s + (8.06 − 7.74i)5-s + (−3.75 + 3.75i)7-s + (24.1 + 11.9i)9-s − 6.48i·11-s + (27.4 + 27.4i)13-s + (49.9 − 29.6i)15-s + (−55.8 − 55.8i)17-s + 10.4i·19-s + (−23.4 + 14.5i)21-s + (−115. + 115. i)23-s + (4.99 − 124. i)25-s + (108. + 89.3i)27-s − 267.·29-s − 305.·31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + (0.721 − 0.692i)5-s + (−0.202 + 0.202i)7-s + (0.895 + 0.444i)9-s − 0.177i·11-s + (0.586 + 0.586i)13-s + (0.860 − 0.509i)15-s + (−0.796 − 0.796i)17-s + 0.126i·19-s + (−0.243 + 0.151i)21-s + (−1.04 + 1.04i)23-s + (0.0399 − 0.999i)25-s + (0.770 + 0.637i)27-s − 1.71·29-s − 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0184i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0184i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.999 + 0.0184i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.00420 - 0.0184978i\)
\(L(\frac12)\) \(\approx\) \(2.00420 - 0.0184978i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.05 - 1.18i)T \)
5 \( 1 + (-8.06 + 7.74i)T \)
good7 \( 1 + (3.75 - 3.75i)T - 343iT^{2} \)
11 \( 1 + 6.48iT - 1.33e3T^{2} \)
13 \( 1 + (-27.4 - 27.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (55.8 + 55.8i)T + 4.91e3iT^{2} \)
19 \( 1 - 10.4iT - 6.85e3T^{2} \)
23 \( 1 + (115. - 115. i)T - 1.21e4iT^{2} \)
29 \( 1 + 267.T + 2.43e4T^{2} \)
31 \( 1 + 305.T + 2.97e4T^{2} \)
37 \( 1 + (-187. + 187. i)T - 5.06e4iT^{2} \)
41 \( 1 + 374. iT - 6.89e4T^{2} \)
43 \( 1 + (-291. - 291. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-230. - 230. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-178. + 178. i)T - 1.48e5iT^{2} \)
59 \( 1 - 225.T + 2.05e5T^{2} \)
61 \( 1 + 315.T + 2.26e5T^{2} \)
67 \( 1 + (-203. + 203. i)T - 3.00e5iT^{2} \)
71 \( 1 - 161. iT - 3.57e5T^{2} \)
73 \( 1 + (-300. - 300. i)T + 3.89e5iT^{2} \)
79 \( 1 + 948. iT - 4.93e5T^{2} \)
83 \( 1 + (553. - 553. i)T - 5.71e5iT^{2} \)
89 \( 1 - 577.T + 7.04e5T^{2} \)
97 \( 1 + (-84.9 + 84.9i)T - 9.12e5iT^{2} \)
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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

−14.37830454771524346765994574472, −13.56248467793877396471855597176, −12.69624490272390843249498149535, −11.05598308571421875691882392779, −9.441016471256401143629238072236, −9.045187447163474348254975627017, −7.51145470279425181636527653625, −5.71786221426095405602814344957, −4.00523781466870002029369537366, −2.02323815934912556461247237368, 2.13177089886048101420317013605, 3.76157364471369755805204728487, 6.08710501101516703577556828153, 7.34043921618211627083905987932, 8.706814928405662176876990717410, 9.893981144305214274041221604383, 10.91934017875734295554074923141, 12.79735075361409108503376522133, 13.47635535101467741628331104739, 14.59518639568838392187594872682

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