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Properties

Label 2-60-60.59-c3-0-21
Degree $2$
Conductor $60$
Sign $0.978 + 0.208i$
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 + (14.5 + 2.16i)6-s − 26.5·7-s + (−2.20 − 22.5i)8-s + (4.37 + 26.6i)9-s + (31.5 + 1.96i)10-s + 17.2·11-s + (38.1 − 16.5i)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 + (0.989 + 0.147i)6-s − 1.43·7-s + (−0.0976 − 0.995i)8-s + (0.162 + 0.986i)9-s + (0.998 + 0.0622i)10-s + 0.471·11-s + (0.917 − 0.397i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.978 + 0.208i$
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: \((2,\ 60,\ (\ :3/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.64586 - 0.278920i\)
\(L(\frac12)\) \(\approx\) \(2.64586 - 0.278920i\)
\(L(\frac{5}{2})\) 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

−14.42744114323288229665410849141, −13.40667625189077223521496854441, −12.75806084230380253138658387166, −10.84640995224778204633442806722, −10.07675107090873692855078141691, −9.162480262560369067385280047023, −6.87488700995247029434088844894, −5.57914665844338855584530445647, −3.66777783162875879457400537449, −2.58702555377734457485957415735, 2.38607515947448607820533813213, 4.14879267005520218638163500653, 6.25312537794803490715039953772, 6.86247779466712095454029744751, 8.714116965303769844127885920471, 9.520692169559671703094418963028, 11.78543593496274077632596491078, 12.92277592787506702182718858293, 13.44012790712809743797332861360, 14.30858989012236455232224991862

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