Properties

Label 2-1050-7.5-c2-0-36
Degree $2$
Conductor $1050$
Sign $0.558 + 0.829i$
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 + (5.26 − 4.61i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (5.41 + 9.37i)11-s + (−2.99 − 1.73i)12-s − 19.2i·13-s + (1.93 + 9.70i)14-s + (−2.00 + 3.46i)16-s + (8.89 − 5.13i)17-s + (2.12 + 3.67i)18-s + (−18.0 − 10.4i)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 + (0.751 − 0.659i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.491 + 0.852i)11-s + (−0.249 − 0.144i)12-s − 1.48i·13-s + (0.137 + 0.693i)14-s + (−0.125 + 0.216i)16-s + (0.523 − 0.302i)17-s + (0.117 + 0.204i)18-s + (−0.951 − 0.549i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.558 + 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.868927454\)
\(L(\frac12)\) \(\approx\) \(1.868927454\)
\(L(2)\) 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 \( 1 + (-5.26 + 4.61i)T \)
good11 \( 1 + (-5.41 - 9.37i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 19.2iT - 169T^{2} \)
17 \( 1 + (-8.89 + 5.13i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (18.0 + 10.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.5 + 18.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 19.0T + 841T^{2} \)
31 \( 1 + (34.6 - 20.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (25.1 - 43.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 22.7iT - 1.68e3T^{2} \)
43 \( 1 - 48.4T + 1.84e3T^{2} \)
47 \( 1 + (57.6 + 33.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-2.47 - 4.28i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (24.4 - 14.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.6 + 35.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.65 - 16.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 49.4T + 5.04e3T^{2} \)
73 \( 1 + (-115. + 66.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-45.0 + 78.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 101. iT - 6.88e3T^{2} \)
89 \( 1 + (-34.3 - 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 68.6iT - 9.40e3T^{2} \)
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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

−9.438375659902204633542168950321, −8.550239000898949138907342905779, −7.939094574907414244087236559495, −7.18822127080248374154043188355, −6.51625656947097895481025536909, −5.22260411962976726604838242162, −4.49922562016579088381535174400, −3.24594950502401660083108739897, −1.84336854533919440983517092078, −0.64004500570792864264895198614, 1.43203077918030866373425686205, 2.29172482623205313381396046389, 3.56640823100554485548931873239, 4.30149528042787568924013297218, 5.44447168833141857684548259507, 6.49611588906431701325542283116, 7.69971264725812959110010255556, 8.420097085260941157108595028553, 9.126698651410750970897937172478, 9.582044219434662221359562402243

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