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Properties

Label 2-1050-35.12-c1-0-15
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: $\chi_{1050} (607, \cdot )$
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

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

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