Properties

Label 2-1110-185.159-c1-0-8
Degree $2$
Conductor $1110$
Sign $0.406 - 0.913i$
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 + (−0.00696 + 2.23i)5-s + 0.999i·6-s + (3.60 + 2.08i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (1.93 − 1.11i)10-s + 5.62·11-s + (0.866 − 0.499i)12-s + (−2.80 + 4.85i)13-s − 4.16i·14-s + (1.12 − 1.93i)15-s + (−0.5 − 0.866i)16-s + (1.43 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.00311 + 0.999i)5-s + 0.408i·6-s + (1.36 + 0.786i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.613 − 0.351i)10-s + 1.69·11-s + (0.249 − 0.144i)12-s + (−0.777 + 1.34i)13-s − 1.11i·14-s + (0.290 − 0.499i)15-s + (−0.125 − 0.216i)16-s + (0.347 + 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.406 - 0.913i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147869173\)
\(L(\frac12)\) \(\approx\) \(1.147869173\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.00696 - 2.23i)T \)
37 \( 1 + (2.01 + 5.73i)T \)
good7 \( 1 + (-3.60 - 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + (2.80 - 4.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.43 - 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.16 + 2.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 - 9.90iT - 29T^{2} \)
31 \( 1 + 4.33iT - 31T^{2} \)
41 \( 1 + (0.470 - 0.815i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.29T + 43T^{2} \)
47 \( 1 + 0.365iT - 47T^{2} \)
53 \( 1 + (3.87 - 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.52 + 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.20 - 1.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.92 - 1.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.08 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 + (8.57 + 4.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.157 + 0.0907i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.38 - 4.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.96T + 97T^{2} \)
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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

−10.12083846926877711238840142059, −9.142413576135216297569865508781, −8.548749023388937385411486319448, −7.45383196010878141055787366962, −6.72867105764554285512272755026, −5.89431475529477387019386931698, −4.60130739777583459947981327757, −3.86109148943630210045453327645, −2.20592195358077259406201889731, −1.70508828136092129350812405116, 0.63857483576614362832513263253, 1.69926449144984937485077906945, 4.02895951426498882965503638098, 4.51348885044129364766473399357, 5.43447106783200435078701799538, 6.21924930384098846036022924086, 7.36692417825053833973973017897, 8.108367057471691246820488727876, 8.646395336098065900847723157151, 9.851277585336465056967528360913

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