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Properties

Label 2-1050-105.89-c1-0-31
Degree $2$
Conductor $1050$
Sign $0.933 + 0.357i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.24 − 1.20i)3-s + (−0.499 − 0.866i)4-s + (0.421 + 1.68i)6-s + (2.64 + 0.0551i)7-s + 0.999·8-s + (0.0969 − 2.99i)9-s + (0.167 − 0.0969i)11-s + (−1.66 − 0.475i)12-s − 1.54·13-s + (−1.37 + 2.26i)14-s + (−0.5 + 0.866i)16-s + (0.458 − 0.264i)17-s + (2.54 + 1.58i)18-s + (5.53 + 3.19i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.718 − 0.695i)3-s + (−0.249 − 0.433i)4-s + (0.171 + 0.685i)6-s + (0.999 + 0.0208i)7-s + 0.353·8-s + (0.0323 − 0.999i)9-s + (0.0506 − 0.0292i)11-s + (−0.480 − 0.137i)12-s − 0.429·13-s + (−0.366 + 0.604i)14-s + (−0.125 + 0.216i)16-s + (0.111 − 0.0642i)17-s + (0.600 + 0.373i)18-s + (1.26 + 0.732i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.933 + 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.890971796\)
\(L(\frac12)\) \(\approx\) \(1.890971796\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.24 + 1.20i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.0551i)T \)
good11 \( 1 + (-0.167 + 0.0969i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (-0.458 + 0.264i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.53 - 3.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.87iT - 29T^{2} \)
31 \( 1 + (-8.02 + 4.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.52 + 0.881i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.91T + 41T^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 + (8.49 + 4.90i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0324 - 0.0562i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.01 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 - 2.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.18 + 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (-4.04 - 7.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.38 + 5.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.11iT - 83T^{2} \)
89 \( 1 + (-8.18 + 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.718879613591582020635611717676, −8.645816333594529643153218633194, −8.291990359078021554992292165018, −7.37925334097441660796815367947, −6.84951889974682482319366379569, −5.67406024962120394894304343846, −4.80771882796768006346418981813, −3.52459088087046018935909597815, −2.20318497967360654238371226497, −1.04297790454112527841880074416, 1.41222139234118802085438834836, 2.64230528185318723945037467819, 3.50759113133628607237559729731, 4.72865045386146961617999669382, 5.14755004850043571227719485542, 6.85746695059448401980166071630, 7.972941040370615843146204155794, 8.231422155298824600219153848142, 9.434222757072570315785814244520, 9.739458875580262214141897947186

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