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Properties

Label 2-60-60.59-c3-0-15
Degree $2$
Conductor $60$
Sign $0.760 - 0.649i$
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  + (2.40 + 1.49i)2-s + (−3.96 − 3.36i)3-s + (3.54 + 7.17i)4-s + (9.10 + 6.48i)5-s + (−4.49 − 13.9i)6-s + 26.5·7-s + (−2.20 + 22.5i)8-s + (4.37 + 26.6i)9-s + (12.1 + 29.1i)10-s − 17.2·11-s + (10.1 − 40.3i)12-s − 66.7i·13-s + (63.7 + 39.6i)14-s + (−14.2 − 56.3i)15-s + (−38.9 + 50.7i)16-s − 87.8·17-s + ⋯
L(s)  = 1  + (0.849 + 0.527i)2-s + (−0.762 − 0.647i)3-s + (0.442 + 0.896i)4-s + (0.814 + 0.579i)5-s + (−0.305 − 0.952i)6-s + 1.43·7-s + (−0.0976 + 0.995i)8-s + (0.162 + 0.986i)9-s + (0.385 + 0.922i)10-s − 0.471·11-s + (0.243 − 0.970i)12-s − 1.42i·13-s + (1.21 + 0.756i)14-s + (−0.245 − 0.969i)15-s + (−0.608 + 0.793i)16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.760 - 0.649i$
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: \((2,\ 60,\ (\ :3/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.97157 + 0.727194i\)
\(L(\frac12)\) \(\approx\) \(1.97157 + 0.727194i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.40 - 1.49i)T \)
3 \( 1 + (3.96 + 3.36i)T \)
5 \( 1 + (-9.10 - 6.48i)T \)
good7 \( 1 - 26.5T + 343T^{2} \)
11 \( 1 + 17.2T + 1.33e3T^{2} \)
13 \( 1 + 66.7iT - 2.19e3T^{2} \)
17 \( 1 + 87.8T + 4.91e3T^{2} \)
19 \( 1 + 22.3iT - 6.85e3T^{2} \)
23 \( 1 + 30.5iT - 1.21e4T^{2} \)
29 \( 1 - 54.3iT - 2.43e4T^{2} \)
31 \( 1 + 143. iT - 2.97e4T^{2} \)
37 \( 1 + 285. iT - 5.06e4T^{2} \)
41 \( 1 - 57.8iT - 6.89e4T^{2} \)
43 \( 1 + 284.T + 7.95e4T^{2} \)
47 \( 1 - 111. iT - 1.03e5T^{2} \)
53 \( 1 - 160.T + 1.48e5T^{2} \)
59 \( 1 + 447.T + 2.05e5T^{2} \)
61 \( 1 - 559.T + 2.26e5T^{2} \)
67 \( 1 + 84.2T + 3.00e5T^{2} \)
71 \( 1 - 119.T + 3.57e5T^{2} \)
73 \( 1 - 253. iT - 3.89e5T^{2} \)
79 \( 1 - 1.18e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 - 859. iT - 7.04e5T^{2} \)
97 \( 1 + 79.1iT - 9.12e5T^{2} \)
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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

−14.60609611710798051620419638109, −13.53216016323756687789119483228, −12.78319781606447102404357180774, −11.33797095806863819805614384024, −10.68896275887066842652147017148, −8.240661595267387829276171802753, −7.17581351026928068939091146281, −5.85135391418506361440641208843, −4.91987189351321600797740859928, −2.28494752736605212692171436371, 1.73794492580093277317687060217, 4.44380992368851840049795700775, 5.15022391508328502031396339157, 6.53020300876242648549278640624, 8.912659344315497222891050145498, 10.17989314536362827736028723155, 11.24796557396376952367949524280, 11.98603440260505104870714178247, 13.38095185764309377676590250030, 14.32703152497418905013750865059

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