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Properties

Label 2-1050-35.12-c1-0-11
Degree $2$
Conductor $1050$
Sign $-0.0677 + 0.997i$
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 + (−1.48 + 2.19i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.366 + 0.633i)11-s + (−0.258 − 0.965i)12-s + (−1.03 − 1.03i)13-s + (0.866 − 2.49i)14-s + (0.500 − 0.866i)16-s + (−2.19 − 0.586i)17-s + (0.965 + 0.258i)18-s + (2.09 − 3.63i)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 + (−0.560 + 0.827i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.110 + 0.191i)11-s + (−0.0747 − 0.278i)12-s + (−0.287 − 0.287i)13-s + (0.231 − 0.668i)14-s + (0.125 − 0.216i)16-s + (−0.531 − 0.142i)17-s + (0.227 + 0.0610i)18-s + (0.481 − 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0677 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0677 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0677 + 0.997i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.0677 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8119709915\)
\(L(\frac12)\) \(\approx\) \(0.8119709915\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 2.19i)T \)
good11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.03 + 1.03i)T + 13iT^{2} \)
17 \( 1 + (2.19 + 0.586i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.09 + 3.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.965 + 3.60i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.19iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.57 + 1.22i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.46iT - 41T^{2} \)
43 \( 1 + (0.138 - 0.138i)T - 43iT^{2} \)
47 \( 1 + (2.84 + 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.08 + 1.36i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.267 - 0.464i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.5 - 6.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 3.86i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (2.07 - 7.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.03 + 4.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.24 + 8.24i)T + 83iT^{2} \)
89 \( 1 + (-3.23 + 5.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.94 + 4.94i)T - 97iT^{2} \)
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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

−9.642419819391661488698900876437, −8.790850509929378847273591396106, −8.151901431989347226175016088909, −7.21803311306713301883937065338, −6.44109980643972388636842819023, −5.73963477115090615666549424939, −4.50418114789493385320363838921, −2.90329746112597205272395879049, −2.19217378699601207557044800962, −0.46878211698380721753043606858, 1.33593446460724260672897268042, 2.91445089024113974155686967740, 3.74861631700029099755606260400, 4.76945012935610625118538926576, 6.03941510490157880694513311125, 6.91798780379424236625811672362, 7.74254694252627626066972661681, 8.614726586217985407247273896496, 9.479856265992487477907321221710, 10.00744340816548673403179230924

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