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Properties

Label 2-1110-185.174-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.346 + 0.937i$
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 + (−1.23 − 1.86i)5-s + 0.999·6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (2 + i)10-s + 4·11-s + (−0.866 + 0.499i)12-s + (−2.59 − 1.5i)13-s − 0.999·14-s + (0.133 + 2.23i)15-s + (−0.5 − 0.866i)16-s + (5.19 − 3i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.550 − 0.834i)5-s + 0.408·6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.632 + 0.316i)10-s + 1.20·11-s + (−0.249 + 0.144i)12-s + (−0.720 − 0.416i)13-s − 0.267·14-s + (0.0345 + 0.576i)15-s + (−0.125 − 0.216i)16-s + (1.26 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.346 + 0.937i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.346 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8900390865\)
\(L(\frac12)\) \(\approx\) \(0.8900390865\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.86i)T \)
37 \( 1 + (-6.06 - 0.5i)T \)
good7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (2.59 + 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + (3.46 - 2i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.7 + 8.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−9.477387311758326732375275561930, −8.930530268342843586913663241597, −7.79824696690196705587251501931, −7.51218649687741737653731357568, −6.37336869617172816832389895257, −5.43116631424145354732702304099, −4.76017925803744806613440611014, −3.47815706567950011829665953626, −1.72096069463101330391663225965, −0.62582294970157489847582728762, 1.17990127193497533532869197485, 2.69436433922611656603532143698, 3.82996987852311543957252923757, 4.53232036040749072216670099539, 6.01064040669676354467895777535, 6.75344901367811815257902473492, 7.55931439571026227098194043731, 8.341247458965401663926363732989, 9.439337457611233151039568004118, 9.996522711310466631563396354719

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