Properties

Label 2-1110-37.27-c1-0-27
Degree $2$
Conductor $1110$
Sign $-0.974 + 0.225i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·6-s + (−1.62 − 2.82i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 5.25·11-s + (0.499 − 0.866i)12-s + (−3.16 + 1.82i)13-s − 3.25i·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (−1.95 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s − 0.408i·6-s + (−0.615 − 1.06i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 1.58·11-s + (0.144 − 0.249i)12-s + (−0.878 + 0.507i)13-s − 0.870i·14-s + (−0.223 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (−0.474 − 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3399101048\)
\(L(\frac12)\) \(\approx\) \(0.3399101048\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (5.27 - 3.03i)T \)
good7 \( 1 + (1.62 + 2.82i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.25T + 11T^{2} \)
13 \( 1 + (3.16 - 1.82i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.21 - 1.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.92iT - 23T^{2} \)
29 \( 1 - 3.37iT - 29T^{2} \)
31 \( 1 + 1.35iT - 31T^{2} \)
41 \( 1 + (-2.73 - 4.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.28iT - 43T^{2} \)
47 \( 1 - 3.32T + 47T^{2} \)
53 \( 1 + (-5.09 + 8.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.55 + 5.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.07 - 0.618i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.93 + 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.215 + 0.372i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.23T + 73T^{2} \)
79 \( 1 + (-6.26 + 3.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.83 + 11.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.75 + 5.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.69iT - 97T^{2} \)
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

−9.571395624690723323424739972205, −8.371894127440851737415979738464, −7.56059519988876059929894695159, −6.88965590649428919228289721063, −6.16046457932923171491741203378, −5.10459587901315978823383895821, −4.46073940919339191227599847734, −3.11599312474072769311880099811, −2.04698492328620201482504693639, −0.11118236243817074867446417338, 2.38541602757994576616269493022, 2.79403280024217220389614673279, 4.15123500333695926115613407553, 5.33294649048868622046110428388, 5.58650517377617925040079476387, 6.61512387105815908715031756291, 7.67523027647590936875549422870, 8.849677923712423129043227108760, 9.548196520867426574026029422053, 10.58235260147038968122355218207

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