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Properties

Label 2-1050-35.19-c2-0-31
Degree $2$
Conductor $1050$
Sign $0.809 + 0.587i$
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 + (0.264 − 6.99i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (2.03 + 3.52i)11-s + (−1.73 + 2.99i)12-s + 11.1·13-s + (−5.26 + 8.38i)14-s + (−2.00 + 3.46i)16-s + (1.44 + 2.50i)17-s + (3.67 − 2.12i)18-s + (−17.9 − 10.3i)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 + (0.0377 − 0.999i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.184 + 0.320i)11-s + (−0.144 + 0.249i)12-s + 0.855·13-s + (−0.376 + 0.598i)14-s + (−0.125 + 0.216i)16-s + (0.0851 + 0.147i)17-s + (0.204 − 0.117i)18-s + (−0.944 − 0.545i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.809 + 0.587i$
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: \((2,\ 1050,\ (\ :1),\ 0.809 + 0.587i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.563623768\)
\(L(\frac12)\) \(\approx\) \(1.563623768\)
\(L(2)\) 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 \( 1 + (-0.264 + 6.99i)T \)
good11 \( 1 + (-2.03 - 3.52i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
17 \( 1 + (-1.44 - 2.50i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (17.9 + 10.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-26.5 - 15.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 5.75T + 841T^{2} \)
31 \( 1 + (-3.32 + 1.91i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.72 + 4.46i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 56.2iT - 1.68e3T^{2} \)
43 \( 1 - 7.16iT - 1.84e3T^{2} \)
47 \( 1 + (-23.8 + 41.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-28.8 + 16.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-38.3 + 22.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.3 + 34.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-79.7 + 46.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.64T + 5.04e3T^{2} \)
73 \( 1 + (-26.2 - 45.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.7 - 46.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 116.T + 6.88e3T^{2} \)
89 \( 1 + (-85.6 - 49.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 132.T + 9.40e3T^{2} \)
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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

−9.614784883715737005526515195271, −8.917381914951958913463342698734, −8.178818408323225121478124151891, −7.23337038695025314487027977523, −6.51949162275231193895877092341, −5.14114309622972876442990541126, −4.07000492656810410898803179737, −3.40261541303017571959425744811, −2.03755480303861409705441982080, −0.72447314226525343035415195014, 1.00669476932606542161572215029, 2.20683721015505321237863129178, 3.25864032247025061185493227894, 4.71310386904604997017860043647, 5.93079017309610443339498781510, 6.36977511554831272045285025938, 7.38848916231963948059377557617, 8.453039781574648102403905277437, 8.663551951077364757658352786043, 9.535379510515237647827948017175

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