Properties

Degree $2$
Conductor $1050$
Sign $0.631 + 0.775i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.618 + 1.61i)3-s + 4-s + (−0.618 − 1.61i)6-s + (0.381 − 2.61i)7-s − 8-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (0.618 + 1.61i)12-s + 1.23·13-s + (−0.381 + 2.61i)14-s + 16-s + 5.23i·17-s + (2.23 − 2.00i)18-s − 8.47i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.356 + 0.934i)3-s + 0.5·4-s + (−0.252 − 0.660i)6-s + (0.144 − 0.989i)7-s − 0.353·8-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (0.178 + 0.467i)12-s + 0.342·13-s + (−0.102 + 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (0.527 − 0.471i)18-s − 1.94i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.631 + 0.775i$
Motivic weight: \(1\)
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050223207\)
\(L(\frac12)\) \(\approx\) \(1.050223207\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.618 - 1.61i)T \)
5 \( 1 \)
7 \( 1 + (-0.381 + 2.61i)T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 5.23iT - 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 + 0.763iT - 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + 4.94iT - 43T^{2} \)
47 \( 1 + 6.47iT - 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 7.23iT - 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 - 0.763T + 97T^{2} \)
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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

−9.882937774708772232176567974628, −8.867300929335106424322617307747, −8.390546487419891105465126420767, −7.58396557850696110996207250030, −6.43890368094671634597026462240, −5.57456832369174855940951363830, −4.26671197495530365215800840917, −3.58333439785814228139229620127, −2.36658827240389304519261698806, −0.57567379642023884554996788649, 1.49295156565076076080166328651, 2.28802839468991303268469151339, 3.42369079984984348551285982004, 5.03474360345807052703714528830, 6.04607466821190160550480782195, 6.82670224911881057051426153471, 7.75027176537115317111508400592, 8.215661967512269195903879676401, 9.255940889839793234875771113376, 9.674960399731557252329355972041

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