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Properties

Label 2-1050-35.12-c1-0-9
Degree $2$
Conductor $1050$
Sign $0.998 + 0.0475i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.998 + 0.0475i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.651848774\)
\(L(\frac12)\) \(\approx\) \(2.651848774\)
\(L(\frac{3}{2})\) 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

−9.609158348919409299576004389203, −9.300021017510689242991131761886, −8.225877139615365878706434232534, −7.15771538715946808932727402325, −6.42797624846516643394887134314, −5.77174210569417948094885550323, −4.58052277203246285973390966302, −3.64246141153497325331940887581, −2.46487036535931724204284371514, −1.51208748549246877187221045458, 1.09805364410008308933405730067, 3.21046462237427380327625326955, 3.51594711833267811916472420854, 4.55563458731447131018573157789, 5.76154905764769097720501712084, 6.20998922819927857823754693488, 7.44413220515906249585655470436, 8.168961246002740127256918041954, 9.070564591425341803219850130585, 10.07737273941556010027086636833

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