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Properties

Label 2-1110-185.159-c1-0-21
Degree $2$
Conductor $1110$
Sign $0.613 - 0.789i$
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 + (−1.38 − 1.75i)5-s + 0.999i·6-s + (1.17 + 0.676i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.831 − 2.07i)10-s + 3.00·11-s + (−0.866 + 0.499i)12-s + (1.12 − 1.94i)13-s + 1.35i·14-s + (−0.317 − 2.21i)15-s + (−0.5 − 0.866i)16-s + (−1.19 − 2.06i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.617 − 0.786i)5-s + 0.408i·6-s + (0.442 + 0.255i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.263 − 0.656i)10-s + 0.905·11-s + (−0.249 + 0.144i)12-s + (0.311 − 0.539i)13-s + 0.361i·14-s + (−0.0819 − 0.571i)15-s + (−0.125 − 0.216i)16-s + (−0.289 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.613 - 0.789i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.302901021\)
\(L(\frac12)\) \(\approx\) \(2.302901021\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 + 1.75i)T \)
37 \( 1 + (3.29 - 5.11i)T \)
good7 \( 1 + (-1.17 - 0.676i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
13 \( 1 + (-1.12 + 1.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.19 + 2.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.57 - 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.77T + 23T^{2} \)
29 \( 1 + 4.07iT - 29T^{2} \)
31 \( 1 - 4.41iT - 31T^{2} \)
41 \( 1 + (-2.30 + 3.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 + (5.39 - 3.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.19 + 2.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.20 - 5.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.08 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.06iT - 73T^{2} \)
79 \( 1 + (-0.509 - 0.294i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.56 - 2.05i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-14.4 + 8.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.59T + 97T^{2} \)
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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

−9.546187771473269986817451329742, −9.034179055103700587671025959135, −8.252972822801371400890812351639, −7.63571639900631296477776098595, −6.69997828983596032912370325792, −5.47203509456286065996903149201, −4.82960751694782923517527684583, −3.91448856651240925000604486214, −3.02744405776166816665776196046, −1.22213332066427113421550295744, 1.15019048671511876823843717451, 2.42830120916690285140819497417, 3.50600801914772859264934553781, 4.11127016243506562119023688108, 5.26271620204051868826402289808, 6.64320253519572802371983641595, 7.06625649731200122120912795918, 8.126697692549564240038902201674, 8.993777598052282336211614008904, 9.698783093643300923143535539767

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