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Properties

Label 2-1050-105.89-c1-0-5
Degree $2$
Conductor $1050$
Sign $-0.994 - 0.107i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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 + (1.73 + 0.0481i)3-s + (−0.499 − 0.866i)4-s + (−0.907 + 1.47i)6-s + (−1.35 + 2.27i)7-s + 0.999·8-s + (2.99 + 0.166i)9-s + (−2.84 + 1.64i)11-s + (−0.824 − 1.52i)12-s − 5.91·13-s + (−1.29 − 2.30i)14-s + (−0.5 + 0.866i)16-s + (−2.08 + 1.20i)17-s + (−1.64 + 2.51i)18-s + (−4.77 − 2.75i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.999 + 0.0277i)3-s + (−0.249 − 0.433i)4-s + (−0.370 + 0.602i)6-s + (−0.511 + 0.859i)7-s + 0.353·8-s + (0.998 + 0.0555i)9-s + (−0.857 + 0.495i)11-s + (−0.237 − 0.439i)12-s − 1.64·13-s + (−0.345 − 0.617i)14-s + (−0.125 + 0.216i)16-s + (−0.504 + 0.291i)17-s + (−0.387 + 0.591i)18-s + (−1.09 − 0.632i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.994 - 0.107i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8256592040\)
\(L(\frac12)\) \(\approx\) \(0.8256592040\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0481i)T \)
5 \( 1 \)
7 \( 1 + (1.35 - 2.27i)T \)
good11 \( 1 + (2.84 - 1.64i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + (2.08 - 1.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.77 + 2.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.80iT - 29T^{2} \)
31 \( 1 + (-4.50 + 2.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.62 - 0.940i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.103T + 41T^{2} \)
43 \( 1 - 1.48iT - 43T^{2} \)
47 \( 1 + (-10.7 - 6.23i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.21 - 2.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.82 + 3.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.3 + 7.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.90 - 1.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-4.05 - 7.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.57 + 7.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.54iT - 83T^{2} \)
89 \( 1 + (-5.62 + 9.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.90T + 97T^{2} \)
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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

−9.893435515463070070932813371777, −9.431016148919811673008378449206, −8.664069918902513451153819308462, −7.81569956354510165912744894875, −7.21044991716914999054970911402, −6.25723946034189912484587904880, −5.12724012881241558574860776847, −4.30538004610043884775115597517, −2.79086790660110039951623719045, −2.10800465772379928097653690548, 0.33349326894029461974542058287, 2.18977591479700926048059197601, 2.84882400250106006915731135261, 4.03289483874559533951700512583, 4.72134203424776721697766726061, 6.35059139378681717596738429793, 7.33950440383041791913974999206, 7.937415667046446497286723582030, 8.745123281632613853867879011826, 9.618906140337446967724128052107

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