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Properties

Label 2-1050-35.17-c1-0-0
Degree $2$
Conductor $1050$
Sign $-0.937 - 0.347i$
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  + (−0.258 + 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s − i·6-s + (−0.189 − 2.63i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.5 + 0.866i)11-s + (0.965 + 0.258i)12-s + (−4.05 − 4.05i)13-s + (2.59 + 0.5i)14-s + (0.500 + 0.866i)16-s + (1.93 + 7.20i)17-s + (0.258 + 0.965i)18-s + (1.13 + 1.96i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s − 0.408i·6-s + (−0.0716 − 0.997i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.150 + 0.261i)11-s + (0.278 + 0.0747i)12-s + (−1.12 − 1.12i)13-s + (0.694 + 0.133i)14-s + (0.125 + 0.216i)16-s + (0.468 + 1.74i)17-s + (0.0610 + 0.227i)18-s + (0.260 + 0.450i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4731970622\)
\(L(\frac12)\) \(\approx\) \(0.4731970622\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.189 + 2.63i)T \)
good11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 + (-1.93 - 7.20i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.13 - 1.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.72 + 1.53i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (-6.46 - 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.01 - 3.79i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.26iT - 41T^{2} \)
43 \( 1 + (6.31 - 6.31i)T - 43iT^{2} \)
47 \( 1 + (-1.15 - 0.309i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.258 + 0.965i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.73 + 9.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.92 - 5.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.93 - 0.517i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.92T + 71T^{2} \)
73 \( 1 + (3.86 - 1.03i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.66 - 1.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.34 - 7.34i)T + 83iT^{2} \)
89 \( 1 + (-3.46 - 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.07 - 2.07i)T - 97iT^{2} \)
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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

−10.29769059458777118050111185294, −9.722750773804600438101558312111, −8.258830865199626395712662441523, −7.85432336668298030721147047515, −6.88458192683373907592111844145, −6.12173434437372514653046951984, −5.18888880439394659097856128980, −4.37438603481597538902016825529, −3.29129735489594627087958617364, −1.35471732561214671786286055638, 0.25378414000441899411867544876, 2.04853582431114159954419276417, 2.83870651920713989607696851904, 4.32277490152149904467670307791, 5.13885909970212006690657158617, 6.00521146988363157825372717571, 7.11123999931430948346139297493, 7.891428737091572713286774040789, 9.043349020616406984375733357713, 9.594579995311982008175465511422

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