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Properties

Label 2-1050-7.3-c2-0-36
Degree $2$
Conductor $1050$
Sign $0.479 + 0.877i$
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  + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s − 2.44i·6-s + (−6.51 + 2.55i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−5.79 + 10.0i)11-s + (2.99 − 1.73i)12-s + 7.86i·13-s + (−7.73 − 6.17i)14-s + (−2.00 − 3.46i)16-s + (−23.9 − 13.8i)17-s + (−2.12 + 3.67i)18-s + (27.2 − 15.7i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s − 0.408i·6-s + (−0.930 + 0.365i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.526 + 0.912i)11-s + (0.249 − 0.144i)12-s + 0.604i·13-s + (−0.552 − 0.440i)14-s + (−0.125 − 0.216i)16-s + (−1.40 − 0.811i)17-s + (−0.117 + 0.204i)18-s + (1.43 − 0.826i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6113811595\)
\(L(\frac12)\) \(\approx\) \(0.6113811595\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (6.51 - 2.55i)T \)
good11 \( 1 + (5.79 - 10.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 7.86iT - 169T^{2} \)
17 \( 1 + (23.9 + 13.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-27.2 + 15.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.07 - 15.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 2.30T + 841T^{2} \)
31 \( 1 + (4.55 + 2.63i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.993 - 1.72i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.1iT - 1.68e3T^{2} \)
43 \( 1 + 49.8T + 1.84e3T^{2} \)
47 \( 1 + (-66.3 + 38.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-28.5 + 49.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (60.9 + 35.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.5 - 33.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.0 + 85.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 34.2T + 5.04e3T^{2} \)
73 \( 1 + (16.8 + 9.74i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (45.2 + 78.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 133. iT - 6.88e3T^{2} \)
89 \( 1 + (9.58 - 5.53i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 72.3iT - 9.40e3T^{2} \)
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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

−9.373655360133926442727681266689, −8.907895618529319969426594114138, −7.41368446266769640788535115767, −7.09925984610399520411024054724, −6.26982580490111350869358914223, −5.25108477077721176918043616918, −4.63907849655796747381619843786, −3.29874560034102630961930669169, −2.17301135544904698203273941430, −0.20913409111151719020922279004, 1.02218906423430618061189031899, 2.74795182668775425207934712697, 3.55228101911063604990239989607, 4.49913367441550454054122149423, 5.62522303166985497434824667834, 6.16187001020397876343104716246, 7.21766755799599392359233895389, 8.374271464196386050934495419839, 9.256767521498097534678170212398, 10.13355057079708492523607405940

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