↑

Properties

Label 2-1110-111.68-c1-0-27
Degree $2$
Conductor $1110$
Sign $0.996 - 0.0872i$
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  + (0.707 − 0.707i)2-s + (1.41 + i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.70 − 0.292i)6-s + 2·7-s + (−0.707 − 0.707i)8-s + (1.00 + 2.82i)9-s + 1.00·10-s + (1.00 − 1.41i)12-s + (1 − i)13-s + (1.41 − 1.41i)14-s + (0.292 + 1.70i)15-s − 1.00·16-s + (4.24 + 4.24i)17-s + (2.70 + 1.29i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.816 + 0.577i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.696 − 0.119i)6-s + 0.755·7-s + (−0.250 − 0.250i)8-s + (0.333 + 0.942i)9-s + 0.316·10-s + (0.288 − 0.408i)12-s + (0.277 − 0.277i)13-s + (0.377 − 0.377i)14-s + (0.0756 + 0.440i)15-s − 0.250·16-s + (1.02 + 1.02i)17-s + (0.638 + 0.304i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.996 - 0.0872i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.996 - 0.0872i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.209956801\)
\(L(\frac12)\) \(\approx\) \(3.209956801\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.41 - i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (1 - 6i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 + (7 + 7i)T + 31iT^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-5.65 - 5.65i)T + 59iT^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - 89iT^{2} \)
97 \( 1 + (-7 + 7i)T - 97iT^{2} \)
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

−10.14116991560248139664009266079, −9.136899735804455290838653728209, −8.210991429022719152470729493684, −7.67200813477317898392618582805, −6.20775614029697427791815305532, −5.47929088624892800989972403653, −4.32228864671638402418084076242, −3.72602480215257132495847754023, −2.55415962484326028358638140084, −1.65005308206818863920438839104, 1.32688143756791562871071688207, 2.53304099209962704796509945757, 3.61885296998006644548725224888, 4.69407833192904972852518071414, 5.55620948851054911034164030240, 6.58754683602131486024868992050, 7.40891472771879649146324224441, 8.024705037590234877826023504104, 8.929753602367932166235388604958, 9.423012086922539346659598510929

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