Properties

Label 2-1110-185.64-c1-0-21
Degree $2$
Conductor $1110$
Sign $0.970 - 0.240i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (2.16 − 0.569i)5-s + 0.999i·6-s + (1.99 − 1.15i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.588 + 2.15i)10-s + 3.60·11-s + (−0.866 − 0.499i)12-s + (2.08 + 3.60i)13-s + 2.30i·14-s + (1.58 − 1.57i)15-s + (−0.5 + 0.866i)16-s + (−1.64 + 2.84i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.967 − 0.254i)5-s + 0.408i·6-s + (0.755 − 0.436i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.186 + 0.682i)10-s + 1.08·11-s + (−0.249 − 0.144i)12-s + (0.577 + 1.00i)13-s + 0.616i·14-s + (0.410 − 0.406i)15-s + (−0.125 + 0.216i)16-s + (−0.398 + 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.970 - 0.240i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.970 - 0.240i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.167709436\)
\(L(\frac12)\) \(\approx\) \(2.167709436\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 0.569i)T \)
37 \( 1 + (4.22 - 4.37i)T \)
good7 \( 1 + (-1.99 + 1.15i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.60T + 11T^{2} \)
13 \( 1 + (-2.08 - 3.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.64 - 2.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.53 - 2.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.748T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + 4.54iT - 31T^{2} \)
41 \( 1 + (1.97 + 3.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 1.55iT - 47T^{2} \)
53 \( 1 + (2.02 + 1.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.1 + 5.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.15 + 1.24i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.87 - 3.97i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.361 - 0.626i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.278iT - 73T^{2} \)
79 \( 1 + (-7.52 + 4.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.53 + 3.19i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.28 + 3.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.584887452458521699493842203924, −8.930541768289101663862049511970, −8.375841684088951932848177982827, −7.42636959488721080739094698117, −6.39667450577065159478751458089, −6.06869443590856024157499915266, −4.61909822874089026753251681695, −3.93282439281401869091879580007, −2.04200586120676781464726722790, −1.35112504949512791486924642288, 1.36131873826446378753001062745, 2.36503407596511908040978326330, 3.30688823510568111116209297251, 4.48666230496618655801605342552, 5.43447506408445673780384153290, 6.47554855832501699703517998521, 7.48235785514924824152638705028, 8.656037932947508902389637956542, 8.924807039752061482538454389596, 9.744305900148636160472479303736

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