Properties

Degree $2$
Conductor $1050$
Sign $-0.0460 + 0.998i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.618 − 1.61i)3-s + 4-s + (0.618 − 1.61i)6-s + (0.381 − 2.61i)7-s + 8-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (0.618 − 1.61i)12-s + 1.23·13-s + (0.381 − 2.61i)14-s + 16-s − 5.23i·17-s + (−2.23 − 2.00i)18-s − 8.47i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.356 − 0.934i)3-s + 0.5·4-s + (0.252 − 0.660i)6-s + (0.144 − 0.989i)7-s + 0.353·8-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (0.178 − 0.467i)12-s + 0.342·13-s + (0.102 − 0.699i)14-s + 0.250·16-s − 1.26i·17-s + (−0.527 − 0.471i)18-s − 1.94i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0460 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0460 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0460 + 0.998i$
Motivic weight: \(1\)
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.0460 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.738780289\)
\(L(\frac12)\) \(\approx\) \(2.738780289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 \)
7 \( 1 + (-0.381 + 2.61i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 5.23iT - 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 + 0.763iT - 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 4.94iT - 43T^{2} \)
47 \( 1 - 6.47iT - 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 - 0.763T + 97T^{2} \)
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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

−9.595439635067450937838775202787, −8.876377585119946391312235864699, −7.61369398972896546991146891092, −7.06110224803891365875955789111, −6.70138906176329705354920358828, −5.20569160946380895495173071772, −4.51672718268051537116152611533, −3.26890809992590836219366957442, −2.31363722480896651569688592899, −0.980090286496742946707337151453, 1.92054078373990136184851912075, 3.21633726460588618721200352800, 3.75557977220978641492751469482, 4.92613237736275825366379992445, 5.86276503884085867126285581071, 6.19351256455667677641751673872, 8.079211251067449796923342831259, 8.302702282120593364786538459486, 9.289202205164036263625394891561, 10.25636842846039878044010349089

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