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Properties

Label 2-1050-35.12-c1-0-7
Degree $2$
Conductor $1050$
Sign $0.943 - 0.332i$
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 + (1.38 − 2.25i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.582 + 1.00i)11-s + (0.258 + 0.965i)12-s + (1.92 + 1.92i)13-s + (−0.756 + 2.53i)14-s + (0.500 − 0.866i)16-s + (0.0209 + 0.00560i)17-s + (0.965 + 0.258i)18-s + (−0.989 + 1.71i)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 + (0.524 − 0.851i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.175 + 0.304i)11-s + (0.0747 + 0.278i)12-s + (0.533 + 0.533i)13-s + (−0.202 + 0.677i)14-s + (0.125 − 0.216i)16-s + (0.00507 + 0.00136i)17-s + (0.227 + 0.0610i)18-s + (−0.226 + 0.393i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.943 - 0.332i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188079014\)
\(L(\frac12)\) \(\approx\) \(1.188079014\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 2.25i)T \)
good11 \( 1 + (-0.582 - 1.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.92 - 1.92i)T + 13iT^{2} \)
17 \( 1 + (-0.0209 - 0.00560i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.989 - 1.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.85 + 6.93i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.60iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.2 + 2.74i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + (-7.87 + 7.87i)T - 43iT^{2} \)
47 \( 1 + (-1.05 - 3.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.82 + 0.757i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.34 - 9.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.76 - 14.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + (0.969 - 3.61i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.39 + 0.805i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.74 - 9.74i)T + 83iT^{2} \)
89 \( 1 + (-1.80 + 3.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.265 + 0.265i)T - 97iT^{2} \)
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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

−10.05456121307051722220328705806, −9.109882590988024017140749055414, −8.400558139418051710555881969102, −7.55868468262921095021890791585, −6.66430375518760297296299028213, −5.84317560010794032114657960521, −4.56596516539057541948768117678, −3.95780904446567459324272684471, −2.38779074402121562680123407817, −0.934437665256411842227607000581, 1.01573838524616260402265369931, 2.21664663921047564920209675654, 3.24764813709739151394490106979, 4.72234160379673250991641937908, 5.90210850656620677765317805818, 6.39629848027134408037993195499, 7.79483817014694281228907502051, 8.042636510508865623525087973724, 9.031480364390182374946934390490, 9.719217731067720896442453193285

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