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Properties

Label 2-1110-185.84-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.942 + 0.333i$
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 + (0.649 + 2.13i)5-s − 0.999·6-s + (0.381 − 0.220i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.507 − 2.17i)10-s − 4.82·11-s + (0.866 + 0.499i)12-s + (5.17 − 2.98i)13-s − 0.440·14-s + (1.63 + 1.52i)15-s + (−0.5 + 0.866i)16-s + (4.52 + 2.61i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.290 + 0.956i)5-s − 0.408·6-s + (0.144 − 0.0832i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.160 − 0.688i)10-s − 1.45·11-s + (0.249 + 0.144i)12-s + (1.43 − 0.829i)13-s − 0.117·14-s + (0.421 + 0.394i)15-s + (−0.125 + 0.216i)16-s + (1.09 + 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.942 + 0.333i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.563450425\)
\(L(\frac12)\) \(\approx\) \(1.563450425\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 - 2.13i)T \)
37 \( 1 + (-4.28 + 4.31i)T \)
good7 \( 1 + (-0.381 + 0.220i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + (-5.17 + 2.98i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.52 - 2.61i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.49 + 4.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.03iT - 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
41 \( 1 + (-2.74 - 4.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.34iT - 43T^{2} \)
47 \( 1 - 4.75iT - 47T^{2} \)
53 \( 1 + (-9.44 - 5.45i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.43 - 7.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.63 - 8.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.414 - 0.239i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.95 + 5.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.65iT - 73T^{2} \)
79 \( 1 + (0.379 + 0.657i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.36 - 1.36i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0225 + 0.0390i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.06060071594947747961448343132, −8.870011730676237071956022658859, −8.057585396893334193231736691403, −7.69833415874311055152705091943, −6.49414800609870150998636344856, −5.85977951099801953138443235478, −4.34772640367175556781765782775, −2.94880508473696274548885986547, −2.67847638128379703335080998726, −1.05521235830517227798950258857, 1.10220117071683205031798139357, 2.32226022790116364651599572158, 3.70109974710706505680373144417, 4.88003743299427270091029827648, 5.59782362178378329886926501124, 6.54318316097962275264727371921, 7.87393256861524336863884922140, 8.258425869976929134168712299197, 8.884540387239087853676497676902, 9.972775583742273462300748523340

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