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Properties

Label 2-60-20.3-c1-0-0
Degree $2$
Conductor $60$
Sign $0.742 - 0.669i$
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  + (−1.41 − 0.0912i)2-s + (−0.707 + 0.707i)3-s + (1.98 + 0.257i)4-s + (1.32 + 1.80i)5-s + (1.06 − 0.933i)6-s + (1.86 + 1.86i)7-s + (−2.77 − 0.544i)8-s − 1.00i·9-s + (−1.69 − 2.66i)10-s − 0.728i·11-s + (−1.58 + 1.22i)12-s + (−3.12 − 3.12i)13-s + (−2.46 − 2.80i)14-s + (−2.20 − 0.342i)15-s + (3.86 + 1.02i)16-s + (1.12 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0645i)2-s + (−0.408 + 0.408i)3-s + (0.991 + 0.128i)4-s + (0.590 + 0.807i)5-s + (0.433 − 0.381i)6-s + (0.705 + 0.705i)7-s + (−0.981 − 0.192i)8-s − 0.333i·9-s + (−0.537 − 0.843i)10-s − 0.219i·11-s + (−0.457 + 0.352i)12-s + (−0.866 − 0.866i)13-s + (−0.658 − 0.749i)14-s + (−0.570 − 0.0885i)15-s + (0.966 + 0.255i)16-s + (0.272 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.742 - 0.669i$
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: \((2,\ 60,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.553334 + 0.212756i\)
\(L(\frac12)\) \(\approx\) \(0.553334 + 0.212756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0912i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.32 - 1.80i)T \)
good7 \( 1 + (-1.86 - 1.86i)T + 7iT^{2} \)
11 \( 1 + 0.728iT - 11T^{2} \)
13 \( 1 + (3.12 + 3.12i)T + 13iT^{2} \)
17 \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + (-5.83 + 5.83i)T - 23iT^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (-3.12 + 3.12i)T - 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (5.10 - 5.10i)T - 43iT^{2} \)
47 \( 1 + (2.09 + 2.09i)T + 47iT^{2} \)
53 \( 1 + (-0.484 - 0.484i)T + 53iT^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (-5.10 - 5.10i)T + 67iT^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \)
79 \( 1 - 7.11T + 79T^{2} \)
83 \( 1 + (3.55 - 3.55i)T - 83iT^{2} \)
89 \( 1 - 1.03iT - 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−15.09425846566368733444020248830, −14.75710515582284237011383365522, −12.69957823024840685239536171254, −11.41324579748295637554493296223, −10.59496208992750199621730919392, −9.592237204499065363021444845433, −8.279486892703712789273985863842, −6.78861442324097595161681779551, −5.43689528568042674304442276403, −2.62726573415258421154532584130, 1.63912336428237542663196072666, 5.00239101024269964312071468942, 6.65568342474225312237821383992, 7.83933882858955501505071224003, 9.121596515668271958104885290191, 10.27664113391821455029778148176, 11.46157618837904673207840832693, 12.51889686939756867239089554308, 13.85607538287597096943185710776, 15.14139950590094773034654773369

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