↑

Properties

Label 2-60-20.7-c1-0-2
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

Related objects

Downloads

Learn more

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: $\chi_{60} (7, \cdot )$
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 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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