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Properties

Label 2-1050-15.2-c1-0-5
Degree $2$
Conductor $1050$
Sign $-0.997 - 0.0738i$
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  + (−0.707 + 0.707i)2-s + (0.799 + 1.53i)3-s − 1.00i·4-s + (−1.65 − 0.521i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−1.72 + 2.45i)9-s + 1.70i·11-s + (1.53 − 0.799i)12-s + (−0.921 + 0.921i)13-s − 1.00·14-s − 1.00·16-s + (−4.76 + 4.76i)17-s + (−0.519 − 2.95i)18-s − 5.94i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.461 + 0.887i)3-s − 0.500i·4-s + (−0.674 − 0.212i)6-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.574 + 0.818i)9-s + 0.514i·11-s + (0.443 − 0.230i)12-s + (−0.255 + 0.255i)13-s − 0.267·14-s − 0.250·16-s + (−1.15 + 1.15i)17-s + (−0.122 − 0.696i)18-s − 1.36i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.997 - 0.0738i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.997 - 0.0738i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.033666241\)
\(L(\frac12)\) \(\approx\) \(1.033666241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.799 - 1.53i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 1.70iT - 11T^{2} \)
13 \( 1 + (0.921 - 0.921i)T - 13iT^{2} \)
17 \( 1 + (4.76 - 4.76i)T - 17iT^{2} \)
19 \( 1 + 5.94iT - 19T^{2} \)
23 \( 1 + (-2.49 - 2.49i)T + 23iT^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 3.40T + 31T^{2} \)
37 \( 1 + (-1.02 - 1.02i)T + 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (8.17 - 8.17i)T - 43iT^{2} \)
47 \( 1 + (0.436 - 0.436i)T - 47iT^{2} \)
53 \( 1 + (6.87 + 6.87i)T + 53iT^{2} \)
59 \( 1 + 0.686T + 59T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 + (1.03 + 1.03i)T + 67iT^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (-4.59 + 4.59i)T - 73iT^{2} \)
79 \( 1 + 7.19iT - 79T^{2} \)
83 \( 1 + (-6.64 - 6.64i)T + 83iT^{2} \)
89 \( 1 - 2.25T + 89T^{2} \)
97 \( 1 + (13.2 + 13.2i)T + 97iT^{2} \)
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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

−10.01794697220761521024643109563, −9.450137886477163636355448498264, −8.667040019000020928921822895800, −8.108718992621019050703096864589, −7.02808744050373979783095014138, −6.18554509064621366325762649543, −4.90460334939743795244324710999, −4.50665691676578615088671761119, −3.04844002993609285468643967725, −1.87332996067805828101616396857, 0.49985075431547244181652761874, 1.84651402869119857066318345976, 2.83778984834012669024273281149, 3.85043696192889743745293618591, 5.17007030993422279740581657421, 6.40736399221463314890403943490, 7.18301080225675292142676893724, 7.947276847681132108792484644167, 8.700063347263782832704770516033, 9.314233889003379695629669044865

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