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Properties

Label 2-60-12.11-c5-0-18
Degree $2$
Conductor $60$
Sign $0.531 - 0.846i$
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  + (5.05 − 2.53i)2-s + (1.23 + 15.5i)3-s + (19.1 − 25.6i)4-s + 25i·5-s + (45.6 + 75.4i)6-s + 220. i·7-s + (31.5 − 178. i)8-s + (−239. + 38.4i)9-s + (63.4 + 126. i)10-s + 292.·11-s + (422. + 265. i)12-s + 145.·13-s + (559. + 1.11e3i)14-s + (−388. + 30.9i)15-s + (−292. − 981. i)16-s + 1.42e3i·17-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)2-s + (0.0793 + 0.996i)3-s + (0.597 − 0.801i)4-s + 0.447i·5-s + (0.518 + 0.855i)6-s + 1.69i·7-s + (0.174 − 0.984i)8-s + (−0.987 + 0.158i)9-s + (0.200 + 0.399i)10-s + 0.728·11-s + (0.846 + 0.531i)12-s + 0.238·13-s + (0.762 + 1.51i)14-s + (−0.445 + 0.0355i)15-s + (−0.285 − 0.958i)16-s + 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.531 - 0.846i$
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: \((2,\ 60,\ (\ :5/2),\ 0.531 - 0.846i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.51092 + 1.38781i\)
\(L(\frac12)\) \(\approx\) \(2.51092 + 1.38781i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.05 + 2.53i)T \)
3 \( 1 + (-1.23 - 15.5i)T \)
5 \( 1 - 25iT \)
good7 \( 1 - 220. iT - 1.68e4T^{2} \)
11 \( 1 - 292.T + 1.61e5T^{2} \)
13 \( 1 - 145.T + 3.71e5T^{2} \)
17 \( 1 - 1.42e3iT - 1.41e6T^{2} \)
19 \( 1 + 337. iT - 2.47e6T^{2} \)
23 \( 1 - 4.03e3T + 6.43e6T^{2} \)
29 \( 1 + 3.76e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.55e3iT - 2.86e7T^{2} \)
37 \( 1 + 8.21e3T + 6.93e7T^{2} \)
41 \( 1 + 5.34e3iT - 1.15e8T^{2} \)
43 \( 1 - 6.08e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.99e3T + 2.29e8T^{2} \)
53 \( 1 + 2.41e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.06e4T + 7.14e8T^{2} \)
61 \( 1 - 1.83e4T + 8.44e8T^{2} \)
67 \( 1 - 1.32e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.99e4T + 1.80e9T^{2} \)
73 \( 1 - 2.81e4T + 2.07e9T^{2} \)
79 \( 1 - 9.12e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.51e4T + 3.93e9T^{2} \)
89 \( 1 + 9.66e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.04e5T + 8.58e9T^{2} \)
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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

−14.66746399277240430815677361517, −13.11730042618689382474523973496, −11.83313147298799329732617258915, −11.10453614821991924897633696482, −9.787211982230321370084555076864, −8.705598523960057485612543773286, −6.31492513708824073040352253191, −5.30064737607436845555677238827, −3.74132170506748168781096119698, −2.38915167030217590390389468555, 1.13929034598768099426071913955, 3.40633628269370823985396980587, 4.98974110724574681735509027833, 6.76855756754563149735915759919, 7.33417779990490994368869211097, 8.778129250169134214783878909563, 10.87259768212476329046231440936, 11.98961533996338156183782183813, 13.08805924356074095192966286873, 13.80963092730563580121779687332

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