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Properties

Label 2-1110-185.64-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.102 - 0.994i$
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 + (−2.21 + 0.317i)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 + (1.75 − 1.38i)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 + (−0.989 + 0.141i)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 + (0.453 − 0.356i)15-s + (−0.125 + 0.216i)16-s + (0.289 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.102 - 0.994i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7597774078\)
\(L(\frac12)\) \(\approx\) \(0.7597774078\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.317i)T \)
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

−9.761122669724068121916014692278, −9.375346853557166845171120675990, −8.243015801719022877842918699219, −7.54863520085386557461237234111, −6.72896186915988267614832194523, −5.93481121843859939615809237698, −4.94305367917554728003473558690, −4.03569508911350551132400032119, −2.96868932419871189754194961360, −0.863566569263280972052474160082, 0.60374010398278921687555459562, 1.94779128565291757312692179935, 3.60158307665331302490461560760, 4.02841011043781729721368268253, 5.29132446567041768686636662996, 6.45627960581285723737066615353, 7.26287632056723394713051025431, 8.011688452348049695694097564999, 8.862733596416779291341406990535, 9.831916857978330436936347503294

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