Properties

Label 2-1110-185.84-c1-0-13
Degree $2$
Conductor $1110$
Sign $-0.157 - 0.987i$
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 + (1.94 + 1.11i)5-s + 0.999·6-s + (−3.70 + 2.13i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.12 + 1.93i)10-s − 5.93·11-s + (0.866 + 0.499i)12-s + (1.08 − 0.625i)13-s − 4.27·14-s + (2.23 − 0.00835i)15-s + (−0.5 + 0.866i)16-s + (6.29 + 3.63i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.867 + 0.496i)5-s + 0.408·6-s + (−1.39 + 0.807i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.355 + 0.611i)10-s − 1.79·11-s + (0.249 + 0.144i)12-s + (0.300 − 0.173i)13-s − 1.14·14-s + (0.577 − 0.00215i)15-s + (−0.125 + 0.216i)16-s + (1.52 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.157 - 0.987i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.424850571\)
\(L(\frac12)\) \(\approx\) \(2.424850571\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.11i)T \)
37 \( 1 + (-3.08 - 5.24i)T \)
good7 \( 1 + (3.70 - 2.13i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.93T + 11T^{2} \)
13 \( 1 + (-1.08 + 0.625i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.29 - 3.63i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.05 - 3.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.89iT - 23T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + 2.68T + 31T^{2} \)
41 \( 1 + (1.33 + 2.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.04iT - 43T^{2} \)
47 \( 1 + 4.62iT - 47T^{2} \)
53 \( 1 + (-1.31 - 0.759i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0555 - 0.0961i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.14 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.44 + 2.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.59 + 6.23i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.36iT - 73T^{2} \)
79 \( 1 + (3.41 + 5.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.46 - 3.72i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.39 + 7.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−10.04043151770930576997957670674, −9.348574301719629041010622883835, −8.206948551835846621888426767749, −7.56301879613905493146473918630, −6.56950293131453510127895181294, −5.62178461977751134140188903750, −5.49973458329251942056962435069, −3.38811770150423383207067036851, −3.12429914559246525761505480827, −1.96771278802493662953419297490, 0.796894668625485580330108958725, 2.66384969829523798734591887919, 3.03850934761798743532379776068, 4.38187002087466746912675472570, 5.21049616492461668701425444419, 6.04209763256619674595493851765, 7.04700749306821885818521328325, 7.923170568688007797200492795213, 9.101859487817602375477379455547, 9.857532684112720299049924728224

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