↑

Properties

Label 2-61-61.60-c3-0-10
Degree $2$
Conductor $61$
Sign $-0.178 + 0.983i$
Analytic cond. $3.59911$
Root an. cond. $1.89713$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.89i·2-s + 2.90·3-s − 7.16·4-s + 19.6·5-s − 11.3i·6-s − 3.96i·7-s − 3.24i·8-s − 18.5·9-s − 76.6i·10-s + 46.4i·11-s − 20.8·12-s − 41.5·13-s − 15.4·14-s + 57.1·15-s − 69.9·16-s + 13.5i·17-s + ⋯
L(s)  = 1  − 1.37i·2-s + 0.558·3-s − 0.895·4-s + 1.76·5-s − 0.769i·6-s − 0.213i·7-s − 0.143i·8-s − 0.687·9-s − 2.42i·10-s + 1.27i·11-s − 0.500·12-s − 0.886·13-s − 0.294·14-s + 0.984·15-s − 1.09·16-s + 0.192i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $-0.178 + 0.983i$
Analytic conductor: \(3.59911\)
Root analytic conductor: \(1.89713\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :3/2),\ -0.178 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.25729 - 1.50634i\)
\(L(\frac12)\) \(\approx\) \(1.25729 - 1.50634i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (85.1 - 468. i)T \)
good2 \( 1 + 3.89iT - 8T^{2} \)
3 \( 1 - 2.90T + 27T^{2} \)
5 \( 1 - 19.6T + 125T^{2} \)
7 \( 1 + 3.96iT - 343T^{2} \)
11 \( 1 - 46.4iT - 1.33e3T^{2} \)
13 \( 1 + 41.5T + 2.19e3T^{2} \)
17 \( 1 - 13.5iT - 4.91e3T^{2} \)
19 \( 1 - 64.7T + 6.85e3T^{2} \)
23 \( 1 + 152. iT - 1.21e4T^{2} \)
29 \( 1 - 262. iT - 2.43e4T^{2} \)
31 \( 1 - 60.2iT - 2.97e4T^{2} \)
37 \( 1 - 274. iT - 5.06e4T^{2} \)
41 \( 1 + 43.3T + 6.89e4T^{2} \)
43 \( 1 + 186. iT - 7.95e4T^{2} \)
47 \( 1 + 124.T + 1.03e5T^{2} \)
53 \( 1 + 615. iT - 1.48e5T^{2} \)
59 \( 1 - 24.3iT - 2.05e5T^{2} \)
67 \( 1 + 879. iT - 3.00e5T^{2} \)
71 \( 1 + 227. iT - 3.57e5T^{2} \)
73 \( 1 + 296.T + 3.89e5T^{2} \)
79 \( 1 + 141. iT - 4.93e5T^{2} \)
83 \( 1 + 941.T + 5.71e5T^{2} \)
89 \( 1 - 544. iT - 7.04e5T^{2} \)
97 \( 1 - 1.70e3T + 9.12e5T^{2} \)
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

−13.99143873526967099679136711625, −12.99967524749965891056794853410, −12.10438193819277427095775248596, −10.50488384626594095833655534721, −9.852149751868284386658910089098, −8.950829620597763015329080643109, −6.83577953784938533662620436318, −5.00452269131395060809077657103, −2.85910242705886598102445732480, −1.81884929741622515171688918385, 2.56312194507767592634553694664, 5.49460531776398326706591513719, 5.99392816700805895143225235565, 7.60230656524632533128921276104, 8.882861665987590779784984491540, 9.669105589900705906037300611108, 11.42578242794619584012729240931, 13.46282520610924144711639488141, 13.91966122325452934723794882278, 14.69133341502598267024553777031

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