Properties

Label 2-1050-105.104-c1-0-4
Degree $2$
Conductor $1050$
Sign $-0.307 - 0.951i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.307 - 0.951i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.307 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4842666906\)
\(L(\frac12)\) \(\approx\) \(0.4842666906\)
\(L(\frac{3}{2})\) 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 \( 1 - 14.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.14947705242145874575983958703, −9.553657358273284381074343483640, −8.734649679809390205269404993625, −7.37415549699592195748590625294, −7.07552388893988077869111669105, −5.99294083407951804045759581300, −4.89886090946972697518736162693, −4.14809229960029952903585559467, −2.83882082295117982151459478679, −1.12018324048485312692200148944, 0.34763995613045536531030086021, 1.96244441624210368675743243250, 2.86829857089241342578859911533, 4.67676941304443642555187551119, 5.63265859002482984259960665048, 6.25712241425748167431156801513, 7.30948559596447910548214697672, 7.85425985668114292972382127050, 8.887232066219535193491669963536, 9.657850001975868934820901535449

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