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Properties

Label 2-1050-7.3-c2-0-27
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 + (−6.16 + 10.6i)11-s + (2.99 − 1.73i)12-s − 7.26i·13-s + (−7.73 − 6.17i)14-s + (−2.00 − 3.46i)16-s + (−8.04 − 4.64i)17-s + (2.12 − 3.67i)18-s + (5.26 − 3.03i)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 + (−0.560 + 0.970i)11-s + (0.249 − 0.144i)12-s − 0.558i·13-s + (−0.552 − 0.440i)14-s + (−0.125 − 0.216i)16-s + (−0.473 − 0.273i)17-s + (0.117 − 0.204i)18-s + (0.276 − 0.159i)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

\(L(\frac{3}{2})\) \(\approx\) \(1.320994997\)
\(L(\frac12)\) \(\approx\) \(1.320994997\)
\(L(2)\) 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 \( 1 + (6.16 - 10.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 7.26iT - 169T^{2} \)
17 \( 1 + (8.04 + 4.64i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.26 + 3.03i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (1.12 + 1.94i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 42.2T + 841T^{2} \)
31 \( 1 + (1.05 + 0.609i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-17.5 - 30.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 57.8iT - 1.68e3T^{2} \)
43 \( 1 - 34.0T + 1.84e3T^{2} \)
47 \( 1 + (49.4 - 28.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-7.27 + 12.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-50.1 - 28.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.07 - 2.93i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.7 + 42.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101.T + 5.04e3T^{2} \)
73 \( 1 + (71.2 + 41.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.8 + 96.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 91.6iT - 6.88e3T^{2} \)
89 \( 1 + (-110. + 63.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 61.4iT - 9.40e3T^{2} \)
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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

−9.916220282630928519491623525449, −8.708847546531546153800120564885, −7.88702448496473099519077765939, −7.31485288755439547829551292299, −6.24090170789060445495969592756, −4.89298030697383340273460208332, −4.55875677242676635233226260188, −2.97505628584380045263197209783, −1.88065139013966097520376689067, −0.74079026477134117188373256348, 0.819935190971333666227009029646, 2.29077891555635451388954538354, 3.87996580647420176401308117056, 4.91322701505258014249734302231, 5.59067285745412870601069260169, 6.40848368629717615992935671646, 7.38553501789975766535714654300, 8.342392646333544864801006357171, 8.802642597091222904578019309967, 9.825818399368273263580840354875

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