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Properties

Label 2-1050-35.12-c1-0-4
Degree $2$
Conductor $1050$
Sign $0.683 - 0.730i$
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 + (−2.63 − 0.189i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.5 − 0.866i)11-s + (0.258 + 0.965i)12-s + (4.05 + 4.05i)13-s + (−2.59 + 0.5i)14-s + (0.500 − 0.866i)16-s + (7.20 + 1.93i)17-s + (−0.965 − 0.258i)18-s + (−1.13 + 1.96i)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 + (−0.997 − 0.0716i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.150 − 0.261i)11-s + (0.0747 + 0.278i)12-s + (1.12 + 1.12i)13-s + (−0.694 + 0.133i)14-s + (0.125 − 0.216i)16-s + (1.74 + 0.468i)17-s + (−0.227 − 0.0610i)18-s + (−0.260 + 0.450i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.683 - 0.730i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.683 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.147473847\)
\(L(\frac12)\) \(\approx\) \(2.147473847\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.05 - 4.05i)T + 13iT^{2} \)
17 \( 1 + (-7.20 - 1.93i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.13 - 1.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.53 - 5.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.79 + 1.01i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.26iT - 41T^{2} \)
43 \( 1 + (6.31 - 6.31i)T - 43iT^{2} \)
47 \( 1 + (-0.309 - 1.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.965 - 0.258i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (-0.517 + 1.93i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.92T + 71T^{2} \)
73 \( 1 + (1.03 - 3.86i)T + (-63.2 - 36.5i)T^{2} \)
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

−10.02119230990346773260674254537, −9.469281321501895970598730534112, −8.457641164132606149298989590427, −7.37541526273759589368463144038, −6.23458138032206308867532245159, −5.87619149973056030158855486930, −4.70227054020907503750791103547, −3.61668030240627334365128543662, −3.20434815413833905363856916221, −1.42310736842252597224300717960, 0.886083143080951814602701437633, 2.70415923326212254429615555512, 3.34474597995016761172799963499, 4.63630208264404902158930100079, 5.77353933133774467372033687658, 6.21196838302840194805326244112, 7.16754770518612856303349821540, 7.978182217761353265759443421789, 8.789536095435537754975266193579, 10.07617923748691752900310602829

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