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Properties

Label 2-1110-185.84-c1-0-17
Degree $2$
Conductor $1110$
Sign $0.216 + 0.976i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.23 + 0.0310i)5-s − 0.999·6-s + (−4.17 + 2.41i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1.95 + 1.09i)10-s + 2.00·11-s + (0.866 + 0.499i)12-s + (1.65 − 0.957i)13-s + 4.82·14-s + (−1.92 + 1.14i)15-s + (−0.5 + 0.866i)16-s + (3.14 + 1.81i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.999 + 0.0138i)5-s − 0.408·6-s + (−1.57 + 0.911i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.617 + 0.345i)10-s + 0.606·11-s + (0.249 + 0.144i)12-s + (0.459 − 0.265i)13-s + 1.28·14-s + (−0.495 + 0.295i)15-s + (−0.125 + 0.216i)16-s + (0.762 + 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.216 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8390351124\)
\(L(\frac12)\) \(\approx\) \(0.8390351124\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (2.23 - 0.0310i)T \)
37 \( 1 + (0.348 + 6.07i)T \)
good7 \( 1 + (4.17 - 2.41i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.00T + 11T^{2} \)
13 \( 1 + (-1.65 + 0.957i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.14 - 1.81i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.31 + 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.73iT - 23T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
31 \( 1 + 5.11T + 31T^{2} \)
41 \( 1 + (2.64 + 4.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + 6.25iT - 47T^{2} \)
53 \( 1 + (1.38 + 0.797i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.98 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.97 - 8.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.48 + 3.74i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.61 - 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.404iT - 73T^{2} \)
79 \( 1 + (-2.99 - 5.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.69 + 3.28i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.63iT - 97T^{2} \)
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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

−9.519713228802684003569028190965, −8.722460214214028671819112984971, −8.415310491641371189409876323997, −7.09047397926883654163301430364, −6.71870881768893202340470145758, −5.53743238065793535074849715840, −3.85588899342258063729115725086, −3.33488031958419575989386641705, −2.31753673814457196376419628702, −0.54198013356947935643084268821, 1.02133206157999726549940601519, 3.01297089673023169791249953653, 3.74056087116367910914622861661, 4.59040209292421001446403262780, 6.28181021840189329218153344024, 6.68831116153360379441777462056, 7.68766816438835261202082547429, 8.329282447123475345288425890405, 9.183636543096372940987239461624, 10.01295308932725913526418081890

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