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Properties

Label 2-1050-35.12-c1-0-1
Degree $2$
Conductor $1050$
Sign $-0.997 - 0.0643i$
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 + (−2.15 + 1.52i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.883 + 1.52i)11-s + (0.258 + 0.965i)12-s + (2.71 + 2.71i)13-s + (1.69 − 2.03i)14-s + (0.500 − 0.866i)16-s + (2.14 + 0.574i)17-s + (0.965 + 0.258i)18-s + (−0.886 + 1.53i)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 + (−0.816 + 0.577i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.266 + 0.461i)11-s + (0.0747 + 0.278i)12-s + (0.752 + 0.752i)13-s + (0.451 − 0.543i)14-s + (0.125 − 0.216i)16-s + (0.520 + 0.139i)17-s + (0.227 + 0.0610i)18-s + (−0.203 + 0.352i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.997 - 0.0643i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.997 - 0.0643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5319887014\)
\(L(\frac12)\) \(\approx\) \(0.5319887014\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 1.52i)T \)
good11 \( 1 + (-0.883 - 1.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.71 - 2.71i)T + 13iT^{2} \)
17 \( 1 + (-2.14 - 0.574i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.886 - 1.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.04 + 3.90i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.84iT - 29T^{2} \)
31 \( 1 + (8.94 - 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.21 - 0.861i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (3.46 - 3.46i)T - 43iT^{2} \)
47 \( 1 + (1.59 + 5.93i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.396 - 0.106i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.18 + 8.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.87 + 3.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.97 - 7.37i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (2.75 - 10.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.9 - 6.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.94 + 1.94i)T + 83iT^{2} \)
89 \( 1 + (-0.558 + 0.966i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.26 - 7.26i)T - 97iT^{2} \)
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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

−10.09520281568108229970597800117, −9.502885497326477067723716393147, −8.818852098800016659007017516611, −8.057041255362103760985154326810, −6.79907358846748818275821360637, −6.27159085271611114624960604039, −5.31918864765868060548348507721, −4.10479866781725391466848115131, −3.08357346149973506566799239894, −1.69554545111732904399777949074, 0.30762193773173996437204896598, 1.56491818397409466991865710587, 3.07816967823989847490338840697, 3.80721399545642164696472420178, 5.52298369433129432094678958528, 6.21085179233242991847117753134, 7.22177246955066181591526757877, 7.70613215200450604927403242184, 8.826600963016515791836271827008, 9.366328361671462562364550229934

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