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Properties

Label 2-1050-35.3-c1-0-3
Degree $2$
Conductor $1050$
Sign $0.153 - 0.988i$
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  + (−0.965 − 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + i·6-s + (0.965 + 2.46i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.92 − 3.33i)11-s + (0.258 − 0.965i)12-s + (−4.42 + 4.42i)13-s + (−0.295 − 2.62i)14-s + (0.500 + 0.866i)16-s + (−1.36 + 0.364i)17-s + (0.965 − 0.258i)18-s + (1.76 + 3.05i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + 0.408i·6-s + (0.365 + 0.930i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.580 − 1.00i)11-s + (0.0747 − 0.278i)12-s + (−1.22 + 1.22i)13-s + (−0.0789 − 0.702i)14-s + (0.125 + 0.216i)16-s + (−0.330 + 0.0884i)17-s + (0.227 − 0.0610i)18-s + (0.404 + 0.700i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7004313780\)
\(L(\frac12)\) \(\approx\) \(0.7004313780\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.965 - 2.46i)T \)
good11 \( 1 + (-1.92 + 3.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.42 - 4.42i)T - 13iT^{2} \)
17 \( 1 + (1.36 - 0.364i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.76 - 3.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.364 - 1.36i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.66iT - 29T^{2} \)
31 \( 1 + (4.55 + 2.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.82 + 2.63i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.741iT - 41T^{2} \)
43 \( 1 + (1.52 + 1.52i)T + 43iT^{2} \)
47 \( 1 + (1.36 - 5.07i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-9.51 + 2.54i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.84 - 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.857 - 0.495i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.98 - 7.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (-2.75 - 10.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.72 - 1.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.471 + 0.471i)T - 83iT^{2} \)
89 \( 1 + (6.97 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.98 - 6.98i)T + 97iT^{2} \)
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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

−9.998172990968187787459984730345, −8.960884854054386005031865773204, −8.764774660473281435809214751855, −7.59940454936367140517726474172, −6.91411492996138237781714134559, −5.97152229985131978596167725884, −5.10941284570970456844290595417, −3.67023560487170384321876668885, −2.40299373497103433902093520410, −1.49463465718163946857547612090, 0.39859934575295350060050044077, 2.04714174375230944595410531866, 3.42631256186414514455553583029, 4.64174086408827383970732115785, 5.24501413760176839759954125124, 6.62553318439829017989504710263, 7.30256839825113533389128917085, 8.000311868356488938247521545950, 9.051795817087082981509994322340, 9.874129714969633111270392540379

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