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Properties

Label 2-1110-185.174-c1-0-23
Degree $2$
Conductor $1110$
Sign $0.834 - 0.551i$
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 + (0.833 − 2.07i)5-s − 0.999·6-s + (3.97 + 2.29i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.315 + 2.21i)10-s + 5.56·11-s + (0.866 − 0.499i)12-s + (3.53 + 2.04i)13-s − 4.59·14-s + (1.75 − 1.38i)15-s + (−0.5 − 0.866i)16-s + (−4.26 + 2.46i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.372 − 0.927i)5-s − 0.408·6-s + (1.50 + 0.867i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.0997 + 0.700i)10-s + 1.67·11-s + (0.249 − 0.144i)12-s + (0.980 + 0.566i)13-s − 1.22·14-s + (0.454 − 0.356i)15-s + (−0.125 − 0.216i)16-s + (−1.03 + 0.597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.834 - 0.551i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.997431115\)
\(L(\frac12)\) \(\approx\) \(1.997431115\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.833 + 2.07i)T \)
37 \( 1 + (-5.45 - 2.68i)T \)
good7 \( 1 + (-3.97 - 2.29i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.56T + 11T^{2} \)
13 \( 1 + (-3.53 - 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.26 - 2.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.172 - 0.298i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + 6.64T + 29T^{2} \)
31 \( 1 + 9.30T + 31T^{2} \)
41 \( 1 + (-1.51 + 2.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.83iT - 43T^{2} \)
47 \( 1 + 0.790iT - 47T^{2} \)
53 \( 1 + (7.59 - 4.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.46 - 4.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.03 + 1.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.44 + 4.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.76 + 9.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.49iT - 73T^{2} \)
79 \( 1 + (4.03 - 6.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.26 + 4.77i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.77 + 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−9.322975416759807648313193434132, −9.031115655633610726428509476412, −8.573585219050854346059760944643, −7.79753062826464402734150676199, −6.49169268405650077885720648367, −5.78908035852156028478293354276, −4.66892524995385166135017030360, −4.02297144364580718304074561999, −2.00938030028007695346306412370, −1.49407016999287695434277063795, 1.29525341750961087753258955878, 2.01046862863204728631131181362, 3.50401684541251822867717503839, 4.10503592964063393034723611555, 5.65706194117551342392963713376, 6.80592795771747082045401310032, 7.34831244927907971035476047600, 8.098229410797676774564721329948, 9.083585432834424721001075145384, 9.578876233567236132205150234265

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