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Properties

Label 2-1110-185.64-c1-0-3
Degree $2$
Conductor $1110$
Sign $-0.895 - 0.444i$
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 + (−2.18 − 0.459i)5-s + 0.999i·6-s + (−1.07 + 0.619i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.49 − 1.66i)10-s + 2.37·11-s + (−0.866 − 0.499i)12-s + (−1.12 − 1.94i)13-s − 1.23i·14-s + (−2.12 + 0.696i)15-s + (−0.5 + 0.866i)16-s + (−0.499 + 0.865i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.978 − 0.205i)5-s + 0.408i·6-s + (−0.405 + 0.233i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.471 − 0.526i)10-s + 0.717·11-s + (−0.249 − 0.144i)12-s + (−0.311 − 0.538i)13-s − 0.330i·14-s + (−0.548 + 0.179i)15-s + (−0.125 + 0.216i)16-s + (−0.121 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.895 - 0.444i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4645155191\)
\(L(\frac12)\) \(\approx\) \(0.4645155191\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.459i)T \)
37 \( 1 + (5.14 - 3.25i)T \)
good7 \( 1 + (1.07 - 0.619i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 + (1.12 + 1.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.499 - 0.865i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.10 - 4.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.09T + 23T^{2} \)
29 \( 1 - 5.72iT - 29T^{2} \)
31 \( 1 - 3.10iT - 31T^{2} \)
41 \( 1 + (-4.36 - 7.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.26T + 43T^{2} \)
47 \( 1 + 6.29iT - 47T^{2} \)
53 \( 1 + (10.0 + 5.82i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.06 - 0.617i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.52 - 2.61i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.63 - 0.944i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.46 - 4.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.123iT - 73T^{2} \)
79 \( 1 + (13.2 - 7.66i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.36 - 3.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.869 - 0.502i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.95T + 97T^{2} \)
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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

−10.04314106989721387229987422450, −9.048159829056688762048191842417, −8.452724401275111905175705537002, −7.88769210696167885519491379064, −6.87676632583183678125492118708, −6.33848493018409740893664868958, −5.05954408553529405013078490024, −4.04854829955169071585980626556, −3.11716889160231318027304031617, −1.50261674838041744581158350003, 0.21817103948750906045951338008, 2.08256536320985959594243156133, 3.16741077302970668252014954258, 4.09726178682183423699157265494, 4.62015070126693601890956169792, 6.39099016202661760251153427257, 7.15203289335500016983943887944, 7.998336247515523341246661183204, 8.892932870433737225299552318900, 9.345972828228651744771787541789

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