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Properties

Label 2-1110-185.84-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.475 - 0.879i$
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 + (−1.88 + 1.20i)5-s − 0.999·6-s + (−0.304 + 0.175i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.23 − 0.106i)10-s − 1.03·11-s + (0.866 + 0.499i)12-s + (1.89 − 1.09i)13-s + 0.351·14-s + (−1.02 + 1.98i)15-s + (−0.5 + 0.866i)16-s + (−4.45 − 2.57i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.841 + 0.540i)5-s − 0.408·6-s + (−0.114 + 0.0663i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.706 − 0.0337i)10-s − 0.312·11-s + (0.249 + 0.144i)12-s + (0.525 − 0.303i)13-s + 0.0938·14-s + (−0.264 + 0.513i)15-s + (−0.125 + 0.216i)16-s + (−1.08 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.475 - 0.879i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9015231578\)
\(L(\frac12)\) \(\approx\) \(0.9015231578\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 - 1.20i)T \)
37 \( 1 + (5.69 - 2.12i)T \)
good7 \( 1 + (0.304 - 0.175i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
13 \( 1 + (-1.89 + 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.45 + 2.57i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.10 - 7.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.23iT - 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 0.148T + 31T^{2} \)
41 \( 1 + (-5.56 - 9.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 6.34iT - 43T^{2} \)
47 \( 1 - 8.96iT - 47T^{2} \)
53 \( 1 + (3.86 + 2.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.49 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.15 - 1.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.4 + 7.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.35 + 9.27i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.52iT - 73T^{2} \)
79 \( 1 + (-4.72 - 8.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.80 + 2.19i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.10 + 3.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−9.885434769109372927466089107954, −9.216469682416184533192060882512, −8.074693783671444025822295521713, −7.87142407993159710126335375789, −6.90448438786389168025592963453, −6.01003052768310589410443413697, −4.51734538389417836555210223676, −3.41862012997675345556449223491, −2.79192123527635200843797658930, −1.32055857964882602183219380580, 0.49273473685545001259728515294, 2.17340065390223957586433992189, 3.49455079328888969373404885664, 4.49366940194712049699793672587, 5.31561925062156822441761262401, 6.69910736074904058276455479762, 7.26308061117359319714167328349, 8.351935747514830320718022720148, 8.738878136348982011508115721091, 9.392186186209400225284104773294

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