↑

Properties

Label 2-1050-5.3-c2-0-1
Degree $2$
Conductor $1050$
Sign $-0.973 - 0.229i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + 2.92·11-s + (−2.44 + 2.44i)12-s + (1.13 + 1.13i)13-s − 3.74i·14-s − 4·16-s + (−1.54 + 1.54i)17-s + (−2.99 − 2.99i)18-s − 3.35i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + 0.265·11-s + (−0.204 + 0.204i)12-s + (0.0871 + 0.0871i)13-s − 0.267i·14-s − 0.250·16-s + (−0.0908 + 0.0908i)17-s + (−0.166 − 0.166i)18-s − 0.176i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2600316182\)
\(L(\frac12)\) \(\approx\) \(0.2600316182\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 - 2.92T + 121T^{2} \)
13 \( 1 + (-1.13 - 1.13i)T + 169iT^{2} \)
17 \( 1 + (1.54 - 1.54i)T - 289iT^{2} \)
19 \( 1 + 3.35iT - 361T^{2} \)
23 \( 1 + (7.90 + 7.90i)T + 529iT^{2} \)
29 \( 1 - 13.5iT - 841T^{2} \)
31 \( 1 - 15.7T + 961T^{2} \)
37 \( 1 + (-16.8 + 16.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 72.6T + 1.68e3T^{2} \)
43 \( 1 + (20.3 + 20.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-52.3 + 52.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (-40.5 - 40.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 117. iT - 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + (57.7 - 57.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 51.7T + 5.04e3T^{2} \)
73 \( 1 + (72.3 + 72.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 37.1iT - 6.24e3T^{2} \)
83 \( 1 + (-21.3 - 21.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (52.5 - 52.5i)T - 9.40e3iT^{2} \)
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

−10.15466334224888654599162456163, −9.057637297288088570450664497738, −8.527337971334264634202312268289, −7.48198047749126092741120049485, −6.79078593279719192970112039875, −6.01790817228890353895816114168, −5.21202277070479779047074088288, −4.05695763706282199065177143494, −2.57828989988562907291425839411, −1.29034883580571915319546056949, 0.10887724806364564108526052772, 1.48835637141135122108096506599, 2.93529251967143498411733825290, 3.89674526820651553616479483299, 4.81383456233066303916315804247, 5.98517985363531516795636367339, 6.81402653835454815251952827566, 7.82022433171282874582714112041, 8.653986675776912180942061376100, 9.588213099978257067925761031724

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