Properties

Label 2-1050-35.19-c2-0-29
Degree $2$
Conductor $1050$
Sign $0.994 - 0.0999i$
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 + (1.64 − 6.80i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (8.87 + 15.3i)11-s + (1.73 − 2.99i)12-s − 2.00·13-s + (6.82 − 7.17i)14-s + (−2.00 + 3.46i)16-s + (−10.6 − 18.3i)17-s + (−3.67 + 2.12i)18-s + (5.70 + 3.29i)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 + (0.234 − 0.972i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.807 + 1.39i)11-s + (0.144 − 0.249i)12-s − 0.154·13-s + (0.487 − 0.512i)14-s + (−0.125 + 0.216i)16-s + (−0.623 − 1.08i)17-s + (−0.204 + 0.117i)18-s + (0.300 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.0999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.0999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.994 - 0.0999i$
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: \((2,\ 1050,\ (\ :1),\ 0.994 - 0.0999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.721204264\)
\(L(\frac12)\) \(\approx\) \(2.721204264\)
\(L(2)\) 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 \( 1 + (-1.64 + 6.80i)T \)
good11 \( 1 + (-8.87 - 15.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.00T + 169T^{2} \)
17 \( 1 + (10.6 + 18.3i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.70 - 3.29i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-34.1 - 19.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 25.1T + 841T^{2} \)
31 \( 1 + (-4.76 + 2.75i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (10.1 + 5.87i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 16.8iT - 1.68e3T^{2} \)
43 \( 1 - 27.7iT - 1.84e3T^{2} \)
47 \( 1 + (-22.8 + 39.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-85.2 + 49.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (10.9 - 6.34i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-82.0 - 47.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (7.23 - 4.17i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 71.8T + 5.04e3T^{2} \)
73 \( 1 + (0.402 + 0.697i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-58.1 + 100. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 50.1T + 6.88e3T^{2} \)
89 \( 1 + (74.9 + 43.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 165.T + 9.40e3T^{2} \)
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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

−9.747780535693584405504727762439, −8.851234562246514670066575217264, −7.62981464925140602421109103261, −7.04917557615359327703253670183, −6.66960492158935673180229094291, −5.24945708248259674363813511997, −4.64751497501481599469275116003, −3.66116537895951163986076978787, −2.28475020651220519895829091455, −0.989276606234792689592418903546, 0.975742677729820273946206820217, 2.50457953171372254478518315826, 3.44367259299852947600221295403, 4.45251401107897254857524160226, 5.35054745053695803344906096304, 6.11000685651499366617277939393, 6.80330063342095575096104009117, 8.462055609790562491845049296875, 8.800284581862192068336083080278, 9.776072861919019989531215386117

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