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Properties

Label 2-60-12.11-c5-0-1
Degree $2$
Conductor $60$
Sign $-0.818 - 0.575i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.72 + 4.25i)2-s + (−11.4 − 10.5i)3-s + (−4.20 − 31.7i)4-s − 25i·5-s + (87.6 − 9.41i)6-s − 94.6i·7-s + (150. + 100. i)8-s + (19.9 + 242. i)9-s + (106. + 93.2i)10-s − 215.·11-s + (−286. + 408. i)12-s − 176.·13-s + (402. + 352. i)14-s + (−264. + 286. i)15-s + (−988. + 266. i)16-s + 0.748i·17-s + ⋯
L(s)  = 1  + (−0.659 + 0.752i)2-s + (−0.735 − 0.677i)3-s + (−0.131 − 0.991i)4-s − 0.447i·5-s + (0.994 − 0.106i)6-s − 0.730i·7-s + (0.832 + 0.554i)8-s + (0.0821 + 0.996i)9-s + (0.336 + 0.294i)10-s − 0.537·11-s + (−0.575 + 0.818i)12-s − 0.290·13-s + (0.549 + 0.481i)14-s + (−0.302 + 0.328i)15-s + (−0.965 + 0.260i)16-s + 0.000628i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.818 - 0.575i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.818 - 0.575i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0460712 + 0.145672i\)
\(L(\frac12)\) \(\approx\) \(0.0460712 + 0.145672i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.72 - 4.25i)T \)
3 \( 1 + (11.4 + 10.5i)T \)
5 \( 1 + 25iT \)
good7 \( 1 + 94.6iT - 1.68e4T^{2} \)
11 \( 1 + 215.T + 1.61e5T^{2} \)
13 \( 1 + 176.T + 3.71e5T^{2} \)
17 \( 1 - 0.748iT - 1.41e6T^{2} \)
19 \( 1 - 1.33e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.64e3T + 6.43e6T^{2} \)
29 \( 1 - 7.26e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.32e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.40e4T + 6.93e7T^{2} \)
41 \( 1 + 1.86e3iT - 1.15e8T^{2} \)
43 \( 1 - 7.40e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.35e4T + 2.29e8T^{2} \)
53 \( 1 + 1.78e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.68e3T + 7.14e8T^{2} \)
61 \( 1 + 4.30e4T + 8.44e8T^{2} \)
67 \( 1 + 1.84e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.42e4T + 1.80e9T^{2} \)
73 \( 1 + 7.44e4T + 2.07e9T^{2} \)
79 \( 1 + 1.32e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.19e4T + 3.93e9T^{2} \)
89 \( 1 - 4.48e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.23e5T + 8.58e9T^{2} \)
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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

−14.52899925350706750578543593750, −13.50569534671635050149380626878, −12.33620117337867242187577095729, −10.89069424334577496758201804383, −9.953160863632240571210542723927, −8.257829193892889967517170009705, −7.34832869451105148545122602334, −6.08797871548252915838818403650, −4.82383727662608058418012171907, −1.44161008983277061621872453531, 0.10187997505592393681439855522, 2.57064754843351398234004263869, 4.31790603786481744809614113043, 6.06258488419651059503554126534, 7.81779697652609025877321064815, 9.339806722100414068803606107402, 10.16700149909139939622811350740, 11.32157710568232657815925352744, 12.00935261537879494602025965312, 13.31854025209624571922254820539

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