↑

Properties

Label 2-1110-185.64-c1-0-20
Degree $2$
Conductor $1110$
Sign $-0.531 + 0.847i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.465 + 2.18i)5-s + 0.999i·6-s + (−3.89 + 2.24i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.66 + 1.49i)10-s + 0.412·11-s + (0.866 + 0.499i)12-s + (−1.10 − 1.90i)13-s + 4.49i·14-s + (−0.690 − 2.12i)15-s + (−0.5 + 0.866i)16-s + (1.63 − 2.82i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.208 + 0.978i)5-s + 0.408i·6-s + (−1.47 + 0.849i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.525 + 0.473i)10-s + 0.124·11-s + (0.249 + 0.144i)12-s + (−0.305 − 0.529i)13-s + 1.20i·14-s + (−0.178 − 0.549i)15-s + (−0.125 + 0.216i)16-s + (0.395 − 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.531 + 0.847i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.531 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5869313394\)
\(L(\frac12)\) \(\approx\) \(0.5869313394\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.465 - 2.18i)T \)
37 \( 1 + (5.42 - 2.74i)T \)
good7 \( 1 + (3.89 - 2.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.412T + 11T^{2} \)
13 \( 1 + (1.10 + 1.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.63 + 2.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 1.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 + 9.16iT - 29T^{2} \)
31 \( 1 + 6.51iT - 31T^{2} \)
41 \( 1 + (1.68 + 2.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 9.32T + 43T^{2} \)
47 \( 1 - 6.71iT - 47T^{2} \)
53 \( 1 + (10.0 + 5.82i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.76 - 2.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.46 + 3.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.71 - 1.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.01 + 1.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.57iT - 73T^{2} \)
79 \( 1 + (7.72 - 4.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.38 + 1.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.49 + 1.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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

−9.747193421488904320123076594968, −9.229003641606287831863476760793, −7.81020626064628188777367292765, −6.79663183129669124254443932926, −6.05832996623631490872801452220, −5.38701583705658048620665188986, −4.07663528903173792106056032931, −3.11970050524882287460559262053, −2.52520612797372886574608972934, −0.26608568670221778351655977221, 1.24813071472892844394455597706, 3.28254870567697993927185544844, 4.08942760821400147312905067687, 5.09930844864079064599248622283, 5.89974464833672277820669861571, 6.89270859520045895097402877525, 7.27191663445345447381059050194, 8.434405103674193087643232717957, 9.223816296040754920370082768571, 10.01627509404186968494119760126

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