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Properties

Label 2-1050-105.104-c1-0-16
Degree $2$
Conductor $1050$
Sign $0.0896 - 0.995i$
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  − 2-s + (1.36 + 1.06i)3-s + 4-s + (−1.36 − 1.06i)6-s + (0.294 + 2.62i)7-s − 8-s + (0.716 + 2.91i)9-s − 1.43i·11-s + (1.36 + 1.06i)12-s + 4.73·13-s + (−0.294 − 2.62i)14-s + 16-s + 2.59i·17-s + (−0.716 − 2.91i)18-s − 2.72i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.786 + 0.616i)3-s + 0.5·4-s + (−0.556 − 0.436i)6-s + (0.111 + 0.993i)7-s − 0.353·8-s + (0.238 + 0.971i)9-s − 0.431i·11-s + (0.393 + 0.308i)12-s + 1.31·13-s + (−0.0786 − 0.702i)14-s + 0.250·16-s + 0.630i·17-s + (−0.168 − 0.686i)18-s − 0.625i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0896 - 0.995i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.0896 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586148038\)
\(L(\frac12)\) \(\approx\) \(1.586148038\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 1.06i)T \)
5 \( 1 \)
7 \( 1 + (-0.294 - 2.62i)T \)
good11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
17 \( 1 - 2.59iT - 17T^{2} \)
19 \( 1 + 2.72iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 5.91iT - 31T^{2} \)
37 \( 1 - 2.39iT - 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 0.432iT - 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 1.05iT - 61T^{2} \)
67 \( 1 + 9.82iT - 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 7.58T + 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 - 3.90T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−9.941199326045672760584235255574, −9.092234981818155872606067066240, −8.546993428648783030868644725650, −8.120797454135003462482337845221, −6.84621612603644066236504737532, −5.90531305409683522622835388904, −4.91697467779775207020266214438, −3.61182027345787771849263636128, −2.78370733915314456408522491765, −1.56950214996938336022135897936, 0.889310530798013407224923511455, 1.93320622404050892587663754646, 3.28891950587675926195582837574, 4.10334736527506044688966067542, 5.64766732751926810226247441558, 6.89523843281334890586365273071, 7.14073208549732333503132731907, 8.236409683673270187857838638490, 8.661463729414643380276327085316, 9.708885631201256572308053402472

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