Properties

Label 2-1110-37.11-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.703 + 0.711i$
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 + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·6-s + (1.08 − 1.87i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 4.64·11-s + (0.499 + 0.866i)12-s + (−4.51 − 2.60i)13-s − 2.16i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (2.73 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (0.408 − 0.707i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 1.39·11-s + (0.144 + 0.249i)12-s + (−1.25 − 0.723i)13-s − 0.577i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.663 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.703 + 0.711i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.703 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.401057015\)
\(L(\frac12)\) \(\approx\) \(2.401057015\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-2.83 - 5.37i)T \)
good7 \( 1 + (-1.08 + 1.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 + (4.51 + 2.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.73 + 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.178 - 0.102i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + 6.44iT - 29T^{2} \)
31 \( 1 + 6.19iT - 31T^{2} \)
41 \( 1 + (-1.61 + 2.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.70iT - 43T^{2} \)
47 \( 1 - 0.834T + 47T^{2} \)
53 \( 1 + (-0.578 - 1.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.94 + 2.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.21 - 5.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.180 + 0.312i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.26 + 3.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.91T + 73T^{2} \)
79 \( 1 + (-2.06 - 1.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.98 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.4 - 7.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.153iT - 97T^{2} \)
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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

−9.809681926770371196047937021194, −9.401706907885463269561339071822, −7.931235075431123347230050652791, −7.13890339189496494026549598261, −6.15851990917107909489687053794, −5.34855939783012311972262742961, −4.43246737360337744002128737726, −3.67457605642131323604930927422, −2.49337603451683394348889396007, −1.01312276703569242280328435318, 1.50658029193892997995906139159, 2.54331548288534698670610974407, 3.95515922411164055587326005326, 4.98282837912838938572348567461, 5.63089379043498011295869673179, 6.65062414500691569110707247103, 7.10592040113169591640275304559, 8.297671079337301209261871756934, 8.983303675776154191114737957564, 9.830325020634276270291662786666

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