Properties

Label 2-1110-185.64-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.967 - 0.252i$
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 + (2.17 + 0.503i)5-s + 0.999i·6-s + (−3.20 + 1.84i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.52 + 1.63i)10-s + 4.44·11-s + (−0.866 − 0.499i)12-s + (−3.09 − 5.36i)13-s − 3.69i·14-s + (2.13 − 0.653i)15-s + (−0.5 + 0.866i)16-s + (1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.974 + 0.225i)5-s + 0.408i·6-s + (−1.20 + 0.698i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.482 + 0.517i)10-s + 1.33·11-s + (−0.249 − 0.144i)12-s + (−0.858 − 1.48i)13-s − 0.987i·14-s + (0.552 − 0.168i)15-s + (−0.125 + 0.216i)16-s + (0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774345183\)
\(L(\frac12)\) \(\approx\) \(1.774345183\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.503i)T \)
37 \( 1 + (1.62 - 5.86i)T \)
good7 \( 1 + (3.20 - 1.84i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.44T + 11T^{2} \)
13 \( 1 + (3.09 + 5.36i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.88 + 2.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.31T + 23T^{2} \)
29 \( 1 - 0.108iT - 29T^{2} \)
31 \( 1 - 3.58iT - 31T^{2} \)
41 \( 1 + (-0.667 - 1.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.49T + 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 + (-7.70 - 4.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.292 + 0.169i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.4 + 6.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 7.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.66 - 4.61i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.67iT - 73T^{2} \)
79 \( 1 + (4.87 - 2.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.20 + 3.00i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.9 + 8.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−9.637566185748953774264121818840, −9.220661207844677789465340781975, −8.341522123022823061905808355728, −7.09630903631340709748412771629, −6.77323878699263196584793830832, −5.74479636563358248772409446534, −5.10205736875354308664708254999, −3.28346541702877938094635223031, −2.66436692723937098449501307926, −1.02406423111633606888891510552, 1.23669090646041699943481260570, 2.34215159361771168568558746023, 3.59267638214787009837529466914, 4.16800808190336528135134763254, 5.51660931416518991793554077448, 6.69422219783861839461726663848, 7.18135671428696103216543392569, 8.569453110102827923878268033098, 9.367285421153761567911805655229, 9.691045099219746134200675091178

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