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Properties

Label 2-1050-35.24-c2-0-44
Degree $2$
Conductor $1050$
Sign $-0.887 + 0.461i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (4.79 − 5.10i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (9.03 − 15.6i)11-s + (−1.73 − 2.99i)12-s − 18.6·13-s + (2.26 − 9.63i)14-s + (−2.00 − 3.46i)16-s + (−0.770 + 1.33i)17-s + (−3.67 − 2.12i)18-s + (−29.4 + 17.0i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (0.684 − 0.728i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.821 − 1.42i)11-s + (−0.144 − 0.249i)12-s − 1.43·13-s + (0.161 − 0.688i)14-s + (−0.125 − 0.216i)16-s + (−0.0453 + 0.0785i)17-s + (−0.204 − 0.117i)18-s + (−1.55 + 0.895i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.722262258\)
\(L(\frac12)\) \(\approx\) \(2.722262258\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-4.79 + 5.10i)T \)
good11 \( 1 + (-9.03 + 15.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 18.6T + 169T^{2} \)
17 \( 1 + (0.770 - 1.33i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (29.4 - 17.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-23.3 + 13.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 16.4T + 841T^{2} \)
31 \( 1 + (24.1 + 13.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-44.7 + 25.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 37.4iT - 1.68e3T^{2} \)
43 \( 1 - 63.6iT - 1.84e3T^{2} \)
47 \( 1 + (14.1 + 24.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.384 - 0.221i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (64.2 + 37.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-91.0 + 52.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.0 - 6.35i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 45.7T + 5.04e3T^{2} \)
73 \( 1 + (-18.2 + 31.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.5 + 115. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 49.9T + 6.88e3T^{2} \)
89 \( 1 + (85.9 - 49.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 150.T + 9.40e3T^{2} \)
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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

−9.393164080197689976544941620143, −8.429390099834832788258242613688, −7.71942877225220178863187185030, −6.70284701519775007842670257040, −6.03398491486072926798988814554, −4.81321400149083550534434475544, −4.02980052435875062958887414320, −2.95679965542938626212397575641, −1.82446834766410329112970544007, −0.62388123583575184739166554829, 1.96488278083352441951062455278, 2.75680447774461346909373937139, 4.28321822162852656803142561760, 4.70687111210921927188792727582, 5.54936687816737083728148179539, 6.86510489528117004876411708862, 7.33264642605352747577444763830, 8.516070302606931747205109749089, 9.148370496310930365179298421245, 9.946879184588144807314400981575

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