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Properties

Label 2-60-20.3-c1-0-1
Degree $2$
Conductor $60$
Sign $0.388 - 0.921i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.760 + 1.19i)2-s + (−0.707 + 0.707i)3-s + (−0.844 + 1.81i)4-s + (0.432 − 2.19i)5-s + (−1.38 − 0.305i)6-s + (0.611 + 0.611i)7-s + (−2.80 + 0.371i)8-s − 1.00i·9-s + (2.94 − 1.15i)10-s − 5.12i·11-s + (−0.685 − 1.87i)12-s + (1.76 + 1.76i)13-s + (−0.264 + 1.19i)14-s + (1.24 + 1.85i)15-s + (−2.57 − 3.06i)16-s + (−3.76 + 3.76i)17-s + ⋯
L(s)  = 1  + (0.537 + 0.843i)2-s + (−0.408 + 0.408i)3-s + (−0.422 + 0.906i)4-s + (0.193 − 0.981i)5-s + (−0.563 − 0.124i)6-s + (0.231 + 0.231i)7-s + (−0.991 + 0.131i)8-s − 0.333i·9-s + (0.931 − 0.364i)10-s − 1.54i·11-s + (−0.197 − 0.542i)12-s + (0.488 + 0.488i)13-s + (−0.0706 + 0.319i)14-s + (0.321 + 0.479i)15-s + (−0.643 − 0.765i)16-s + (−0.912 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816887 + 0.542405i\)
\(L(\frac12)\) \(\approx\) \(0.816887 + 0.542405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.760 - 1.19i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.432 + 2.19i)T \)
good7 \( 1 + (-0.611 - 0.611i)T + 7iT^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
13 \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 17iT^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + (1.07 - 1.07i)T - 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + (1.76 - 1.76i)T - 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (-6.20 + 6.20i)T - 43iT^{2} \)
47 \( 1 + (-2.29 - 2.29i)T + 47iT^{2} \)
53 \( 1 + (2.62 + 2.62i)T + 53iT^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (6.20 + 6.20i)T + 67iT^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 + (2.25 + 2.25i)T + 73iT^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + (7.95 - 7.95i)T - 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (-0.793 + 0.793i)T - 97iT^{2} \)
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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

−15.59694583664725787108866621589, −14.18035409831292077873410625601, −13.26963365600892501264852028795, −12.16600459989127711208812793503, −10.96740143964805116278765348797, −9.006955749394184117341291336493, −8.375399677077380439041684148243, −6.35288217453026495508372911131, −5.35152973487716363715185917574, −3.96505536266266673223687962790, 2.35091786896788376344780558035, 4.42259064662844709064730415666, 6.09108251466704909981733752467, 7.37919395256781935822675949801, 9.529215513739091676208992147090, 10.63090667725657982010644831924, 11.46285187000426744942851353983, 12.64861741077386589379350872286, 13.62429323517206390598727843356, 14.67955295334597713600709505424

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