Properties

Label 2-1110-185.84-c1-0-31
Degree $2$
Conductor $1110$
Sign $-0.780 + 0.624i$
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 + (−0.00835 + 2.23i)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 + (−1.11 − 1.94i)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 + (−0.00373 + 0.999i)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 + (−0.286 − 0.501i)15-s + (−0.125 + 0.216i)16-s + (−1.52 − 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.780 + 0.624i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2718654308\)
\(L(\frac12)\) \(\approx\) \(0.2718654308\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.00835 - 2.23i)T \)
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

−9.853032641536963438259851320034, −8.660977747829074815236978350864, −7.72243022036159628908629006025, −7.33390942685054171844750660448, −6.28940987151410724079373550288, −5.00479985032515606659913543708, −4.37314732917234936573148501659, −2.94185250499914358903975283063, −1.98419624237586490234134236314, −0.15015010521590743300248589824, 1.51470757520689783734588024044, 2.43334125978356488412415408936, 4.55583394923066803760926963844, 5.28085159709820497735016803749, 5.63332559909891376385127404006, 7.04916004483097611277699505883, 7.897504931862920170968801560502, 8.426258139334669023029073713245, 9.064345223193622335151315669842, 10.18960557505509934392857055643

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