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Properties

Label 2-60-60.47-c2-0-9
Degree $2$
Conductor $60$
Sign $0.962 + 0.272i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.961 + 1.75i)2-s + (−0.903 − 2.86i)3-s + (−2.15 − 3.37i)4-s + (4.95 + 0.663i)5-s + (5.88 + 1.16i)6-s + (7.30 − 7.30i)7-s + (7.98 − 0.535i)8-s + (−7.36 + 5.17i)9-s + (−5.92 + 8.05i)10-s − 4.41·11-s + (−7.69 + 9.20i)12-s + (7.53 − 7.53i)13-s + (5.78 + 19.8i)14-s + (−2.58 − 14.7i)15-s + (−6.73 + 14.5i)16-s + (−0.350 + 0.350i)17-s + ⋯
L(s)  = 1  + (−0.480 + 0.876i)2-s + (−0.301 − 0.953i)3-s + (−0.538 − 0.842i)4-s + (0.991 + 0.132i)5-s + (0.981 + 0.193i)6-s + (1.04 − 1.04i)7-s + (0.997 − 0.0669i)8-s + (−0.818 + 0.574i)9-s + (−0.592 + 0.805i)10-s − 0.401·11-s + (−0.641 + 0.767i)12-s + (0.579 − 0.579i)13-s + (0.413 + 1.41i)14-s + (−0.172 − 0.985i)15-s + (−0.420 + 0.907i)16-s + (−0.0206 + 0.0206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.978417 - 0.135836i\)
\(L(\frac12)\) \(\approx\) \(0.978417 - 0.135836i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.961 - 1.75i)T \)
3 \( 1 + (0.903 + 2.86i)T \)
5 \( 1 + (-4.95 - 0.663i)T \)
good7 \( 1 + (-7.30 + 7.30i)T - 49iT^{2} \)
11 \( 1 + 4.41T + 121T^{2} \)
13 \( 1 + (-7.53 + 7.53i)T - 169iT^{2} \)
17 \( 1 + (0.350 - 0.350i)T - 289iT^{2} \)
19 \( 1 + 9.24T + 361T^{2} \)
23 \( 1 + (17.9 - 17.9i)T - 529iT^{2} \)
29 \( 1 - 5.52T + 841T^{2} \)
31 \( 1 - 48.1iT - 961T^{2} \)
37 \( 1 + (-3.39 - 3.39i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.0iT - 1.68e3T^{2} \)
43 \( 1 + (1.45 + 1.45i)T + 1.84e3iT^{2} \)
47 \( 1 + (-27.8 - 27.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (52.6 + 52.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 46.1T + 3.72e3T^{2} \)
67 \( 1 + (-32.1 + 32.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 + (72.2 - 72.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 55.9T + 6.24e3T^{2} \)
83 \( 1 + (-46.5 + 46.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 33.2T + 7.92e3T^{2} \)
97 \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \)
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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

−14.52556899160694773945540077895, −13.84414653283558599909675664610, −13.02063083630561781563609299863, −11.06996904441186685804884809498, −10.22556065808638051839526529294, −8.478082142333528372387665906529, −7.51333749303885209990941398954, −6.31462848256407910014898031318, −5.10936558978106438803277912684, −1.42585557678746198378128075014, 2.27905741880963602043412189585, 4.47082931256836556585738941293, 5.80504050437071326933843693110, 8.407174604329641456911771723372, 9.199017927279326004141581795953, 10.34014597532445415852460401073, 11.28568302094546228392992793497, 12.29447630496207704299274789128, 13.73830098299583559444540424410, 14.84620450615413287860040821833

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