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Properties

Label 2-1050-7.5-c2-0-17
Degree $2$
Conductor $1050$
Sign $0.865 - 0.501i$
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 + (−4.24 − 5.56i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (5.42 + 9.40i)11-s + (−2.99 − 1.73i)12-s + 0.772i·13-s + (9.81 − 1.26i)14-s + (−2.00 + 3.46i)16-s + (−16.7 + 9.68i)17-s + (2.12 + 3.67i)18-s + (22.5 + 13.0i)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 + (−0.606 − 0.795i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.493 + 0.854i)11-s + (−0.249 − 0.144i)12-s + 0.0593i·13-s + (0.701 − 0.0902i)14-s + (−0.125 + 0.216i)16-s + (−0.986 + 0.569i)17-s + (0.117 + 0.204i)18-s + (1.18 + 0.686i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.865 - 0.501i$
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: \((2,\ 1050,\ (\ :1),\ 0.865 - 0.501i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.715675507\)
\(L(\frac12)\) \(\approx\) \(1.715675507\)
\(L(2)\) 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 \( 1 + (4.24 + 5.56i)T \)
good11 \( 1 + (-5.42 - 9.40i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 0.772iT - 169T^{2} \)
17 \( 1 + (16.7 - 9.68i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-22.5 - 13.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (6.84 - 11.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 6.99T + 841T^{2} \)
31 \( 1 + (-22.7 + 13.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-32.3 + 55.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 5.54iT - 1.68e3T^{2} \)
43 \( 1 - 68.9T + 1.84e3T^{2} \)
47 \( 1 + (-19.5 - 11.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-37.2 - 64.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-96.6 + 55.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.9 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.1 + 38.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 31.9T + 5.04e3T^{2} \)
73 \( 1 + (-92.6 + 53.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 15.8iT - 6.88e3T^{2} \)
89 \( 1 + (31.8 + 18.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 134. iT - 9.40e3T^{2} \)
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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

−9.527015666722443638159860421090, −9.114876997636401701353885312352, −7.85388631167826029196601529479, −7.42633272423249368035698337393, −6.59238069664018398976606878872, −5.80130408777488998849717979712, −4.39793030412548804482131694766, −3.70018731426622838283713024265, −2.19351591456007172231654845696, −0.899000252324937275840868011647, 0.78339260194361252421490283975, 2.43836587181436687110706268202, 3.05230655885445299552870529216, 4.11683703978269769611623707273, 5.21267380284671713462610839837, 6.30988212657697636254705172205, 7.24737013905544702609495904653, 8.425889334001525271895280059680, 8.900668193948156716623583937559, 9.573193759855994392212770846929

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