↑

Properties

Label 2-61-61.13-c1-0-1
Degree $2$
Conductor $61$
Sign $0.199 - 0.979i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (−0.882 + 1.52i)2-s + 1.30·3-s + (−0.555 − 0.962i)4-s + (−0.184 + 0.320i)5-s + (−1.15 + 1.99i)6-s + (0.283 − 0.490i)7-s − 1.56·8-s − 1.28·9-s + (−0.326 − 0.564i)10-s + 1.67·11-s + (−0.727 − 1.26i)12-s + (2.53 − 4.39i)13-s + (0.5 + 0.866i)14-s + (−0.241 + 0.419i)15-s + (2.49 − 4.31i)16-s + (0.270 + 0.468i)17-s + ⋯
L(s)  = 1  + (−0.623 + 1.08i)2-s + 0.755·3-s + (−0.277 − 0.481i)4-s + (−0.0826 + 0.143i)5-s + (−0.471 + 0.816i)6-s + (0.107 − 0.185i)7-s − 0.553·8-s − 0.428·9-s + (−0.103 − 0.178i)10-s + 0.506·11-s + (−0.210 − 0.363i)12-s + (0.703 − 1.21i)13-s + (0.133 + 0.231i)14-s + (−0.0624 + 0.108i)15-s + (0.623 − 1.07i)16-s + (0.0655 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.199 - 0.979i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.199 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611640 + 0.499770i\)
\(L(\frac12)\) \(\approx\) \(0.611640 + 0.499770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-7.68 - 1.39i)T \)
good2 \( 1 + (0.882 - 1.52i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 1.30T + 3T^{2} \)
5 \( 1 + (0.184 - 0.320i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.283 + 0.490i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 + (-2.53 + 4.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.270 - 0.468i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.29 + 2.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 + (-2.08 - 3.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.58 - 2.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 + (1.46 - 2.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.09 + 8.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + (4.07 - 7.05i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (6.32 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.87 + 10.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.450 + 0.780i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.44 + 4.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.465 + 0.806i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.83T + 89T^{2} \)
97 \( 1 + (-4.67 - 8.10i)T + (-48.5 + 84.0i)T^{2} \)
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

−15.30473682717162644338686952934, −14.57404557621257121540942881360, −13.48084466210341047619458529710, −11.96960399188801853622770684896, −10.43699089651159325903045316158, −8.918336008862337925535421830240, −8.288264875921099068665281910221, −7.10792574099043465570228082614, −5.72775112896555338454034893681, −3.29779008396641952806051999461, 2.10132972058016144547268794706, 3.80518369583347500626518781798, 6.23426103615287216641054873392, 8.316104283456485696046221206611, 9.021126439727258394990034094315, 10.14533443210564515548890766300, 11.47774895550021505272255264002, 12.16651508760744891913028241294, 13.77780200313525348755554317792, 14.58985033934184370112748588799

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