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Properties

Label 2-1050-35.12-c1-0-5
Degree $2$
Conductor $1050$
Sign $0.399 - 0.916i$
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 + (0.703 + 2.55i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.989 + 1.71i)11-s + (0.258 + 0.965i)12-s + (2.19 + 2.19i)13-s + (1.33 + 2.28i)14-s + (0.500 − 0.866i)16-s + (−4.44 − 1.19i)17-s + (−0.965 − 0.258i)18-s + (2.10 − 3.65i)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 + (0.265 + 0.963i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.298 + 0.516i)11-s + (0.0747 + 0.278i)12-s + (0.608 + 0.608i)13-s + (0.358 + 0.609i)14-s + (0.125 − 0.216i)16-s + (−1.07 − 0.288i)17-s + (−0.227 − 0.0610i)18-s + (0.483 − 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.399 - 0.916i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319597696\)
\(L(\frac12)\) \(\approx\) \(2.319597696\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.703 - 2.55i)T \)
good11 \( 1 + (-0.989 - 1.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.19 - 2.19i)T + 13iT^{2} \)
17 \( 1 + (4.44 + 1.19i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.10 + 3.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.52 - 5.68i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 + (1.50 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.67 - 0.717i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + (-6.33 + 6.33i)T - 43iT^{2} \)
47 \( 1 + (1.57 + 5.87i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-11.0 - 2.95i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.10 + 3.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.63 - 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.42 - 5.32i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (-3.93 + 14.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.21 - 1.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.52 + 9.52i)T + 83iT^{2} \)
89 \( 1 + (-3.09 + 5.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.48 + 1.48i)T - 97iT^{2} \)
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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

−10.15639895501908088887480083778, −8.994477469583663349735452019916, −8.923663005723945037426667981460, −7.30977569342567100313244035784, −6.58156616314252287540604342023, −5.49811554109864311968391511482, −4.91503983636788963311167041878, −3.94402516173448892356782238989, −2.88244649523822202900662014014, −1.70410119386420476985200082409, 0.896249027067994231401177552782, 2.34518157997736521246052547911, 3.66941221664035415612718704412, 4.39379525726243449300378830342, 5.61590951641789671969415531794, 6.32570927155955494467529197023, 7.11363418879691351851333263103, 7.993268089799371483381947346676, 8.606957882131104994785828978798, 9.938274632888608260635885638333

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