Properties

Label 2-1110-185.159-c1-0-2
Degree $2$
Conductor $1110$
Sign $-0.380 - 0.924i$
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 + (−0.491 + 2.18i)5-s + 0.999i·6-s + (−0.916 − 0.529i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (2.13 − 0.664i)10-s − 0.825·11-s + (0.866 − 0.499i)12-s + (1.81 − 3.14i)13-s + 1.05i·14-s + (1.51 − 1.64i)15-s + (−0.5 − 0.866i)16-s + (−0.742 − 1.28i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.219 + 0.975i)5-s + 0.408i·6-s + (−0.346 − 0.200i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.675 − 0.210i)10-s − 0.248·11-s + (0.249 − 0.144i)12-s + (0.504 − 0.873i)13-s + 0.282i·14-s + (0.391 − 0.424i)15-s + (−0.125 − 0.216i)16-s + (−0.179 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.380 - 0.924i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3146546527\)
\(L(\frac12)\) \(\approx\) \(0.3146546527\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.491 - 2.18i)T \)
37 \( 1 + (5.95 + 1.24i)T \)
good7 \( 1 + (0.916 + 0.529i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.825T + 11T^{2} \)
13 \( 1 + (-1.81 + 3.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.742 + 1.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.75 - 2.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 - 2.02iT - 29T^{2} \)
31 \( 1 - 6.27iT - 31T^{2} \)
41 \( 1 + (1.42 - 2.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + (8.00 - 4.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.33 - 3.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.60 + 3.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.49i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.701 - 1.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.82iT - 73T^{2} \)
79 \( 1 + (3.10 + 1.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.4 - 7.74i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.86 - 1.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.18T + 97T^{2} \)
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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

−10.27974996691340714455388051908, −9.588041254772642593988318160866, −8.350699870042752951730594052312, −7.64336607012408117798539339735, −6.86393245241146725078437759386, −5.97850687253864224195657656784, −4.93854700125065592615118695317, −3.53601329609954516396231721208, −2.96391697417696267698893358301, −1.46321153945933637419821022594, 0.17363405708796233750335647578, 1.69147855454952929946713738084, 3.58429485466107062207766322916, 4.59992454524907753605871579510, 5.33948872827656287177369982595, 6.18581945992860366225680930425, 7.03101849593370856563247403548, 8.053048542683021374793116743395, 8.778035545665449286518986674302, 9.511395291931700330689911615638

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