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Properties

Label 2-1050-35.19-c2-0-41
Degree $2$
Conductor $1050$
Sign $-0.744 + 0.667i$
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 + (6.78 + 1.72i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−9.79 − 16.9i)11-s + (−1.73 + 2.99i)12-s + 2.16·13-s + (−7.08 − 6.91i)14-s + (−2.00 + 3.46i)16-s + (−9.81 − 16.9i)17-s + (3.67 − 2.12i)18-s + (19.8 + 11.4i)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 + (0.969 + 0.246i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.890 − 1.54i)11-s + (−0.144 + 0.249i)12-s + 0.166·13-s + (−0.506 − 0.493i)14-s + (−0.125 + 0.216i)16-s + (−0.577 − 0.999i)17-s + (0.204 − 0.117i)18-s + (1.04 + 0.602i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6351612695\)
\(L(\frac12)\) \(\approx\) \(0.6351612695\)
\(L(2)\) 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 \( 1 + (-6.78 - 1.72i)T \)
good11 \( 1 + (9.79 + 16.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 2.16T + 169T^{2} \)
17 \( 1 + (9.81 + 16.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-19.8 - 11.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (28.1 + 16.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 27.6T + 841T^{2} \)
31 \( 1 + (36.3 - 21.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (52.3 + 30.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 66.3iT - 1.68e3T^{2} \)
43 \( 1 + 73.9iT - 1.84e3T^{2} \)
47 \( 1 + (30.4 - 52.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-80.2 + 46.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-24.9 + 14.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (72.7 + 42.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.81 - 1.62i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 38.9T + 5.04e3T^{2} \)
73 \( 1 + (-17.6 - 30.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.7 + 18.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 40.3T + 6.88e3T^{2} \)
89 \( 1 + (42.1 + 24.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 25.2T + 9.40e3T^{2} \)
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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

−9.312988303469596737085895912004, −8.592414131882862784529763478103, −8.051943551326857558014360064823, −7.21844580471996186868079238524, −5.76439231481264159919206287051, −5.13100658614041421937425340742, −3.83034175968421382945490831020, −2.92242997884798800796139157681, −1.80464971965855884609749615344, −0.21753669469347886340368660546, 1.56916705513928930912937570746, 2.21206511902679146314107897938, 3.88180321717415576902099386228, 5.01065085347987170601222125323, 5.82752612393076537876276947666, 7.19111951824973219045443757490, 7.44030482839040006246657073005, 8.246353211998582172511422794405, 9.086950603975141218073330002211, 9.978570892846804868628095786676

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