Properties

Degree $2$
Conductor $1050$
Sign $-0.192 - 0.981i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (1.73 + 2i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (2.5 + 4.33i)11-s + (−0.866 − 0.499i)12-s − 5i·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + (0.866 − 0.499i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (0.654 + 0.755i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.753 + 1.30i)11-s + (−0.249 − 0.144i)12-s − 1.38i·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + (0.204 − 0.117i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.192 - 0.981i$
Motivic weight: \(1\)
Character: $\chi_{1050} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.080612481\)
\(L(\frac12)\) \(\approx\) \(2.080612481\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 - 2i)T \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + (9.52 + 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−10.13206576430589535984812050184, −9.457899527853407590647398402883, −8.203942943337333692580195073412, −7.71362690989653334218789450068, −6.52975980915240054040325065528, −5.77387077129675827100604285956, −5.01310478053945939279262613111, −4.22769619086154226136444841032, −3.04098226914475737464000653999, −1.64621132292252629200048457183, 0.881726754073095681531316022913, 2.03869198244360550765590247557, 3.59178548527415140290518200655, 4.34395517083850080423882480809, 5.26837170203719485089935673111, 6.35651595112453648970859603778, 6.82497705710978033604737071180, 7.965523070310263128836411578620, 8.881109539428411400235510862392, 9.836230913242713066857409131356

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