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Properties

Label 2-1110-185.159-c1-0-17
Degree $2$
Conductor $1110$
Sign $-0.521 - 0.853i$
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 + (−1.19 + 1.88i)5-s + 0.999i·6-s + (3.17 + 1.83i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−2.23 − 0.0953i)10-s + 4.74·11-s + (−0.866 + 0.499i)12-s + (1.59 − 2.76i)13-s + 3.66i·14-s + (−1.98 + 1.03i)15-s + (−0.5 − 0.866i)16-s + (2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.536 + 0.843i)5-s + 0.408i·6-s + (1.19 + 0.692i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.706 − 0.0301i)10-s + 1.43·11-s + (−0.249 + 0.144i)12-s + (0.442 − 0.766i)13-s + 0.979i·14-s + (−0.511 + 0.267i)15-s + (−0.125 − 0.216i)16-s + (0.725 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.499028110\)
\(L(\frac12)\) \(\approx\) \(2.499028110\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.88i)T \)
37 \( 1 + (5.81 - 1.77i)T \)
good7 \( 1 + (-3.17 - 1.83i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 + (-1.59 + 2.76i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.87 + 2.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 + 3.80iT - 29T^{2} \)
31 \( 1 - 6.38iT - 31T^{2} \)
41 \( 1 + (-0.175 + 0.304i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-6.59 + 3.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.74 - 5.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.3 + 7.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.485 - 0.280i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.31 - 10.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.67iT - 73T^{2} \)
79 \( 1 + (-1.65 - 0.955i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.66 + 2.69i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-10.8 + 6.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.07T + 97T^{2} \)
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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

−10.22546836964872702697456163528, −8.794213106563775300771294351659, −8.532003529716153903441462376945, −7.72399973475909294665343323395, −6.75352268923205940753966719648, −5.98704496602075919346831540169, −4.88915652047830323931534287176, −3.95840778258948484792321633864, −3.20931925348125298831654181323, −1.80872095079625016628138954638, 1.08349702350299450917940506969, 1.77597214363280672539049424860, 3.46963651461615655173284338759, 4.26428986263800608084221493270, 4.81076150097099572492132286396, 6.13702255584799549816458152265, 7.25438664892841035218275295688, 7.967584696411725999087599950868, 8.962499110591246193911353368230, 9.274601974736470783763307030330

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