Properties

Label 2-1050-35.33-c1-0-2
Degree $2$
Conductor $1050$
Sign $-0.991 + 0.126i$
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 + (−2.52 − 0.781i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−1.31 − 2.27i)11-s + (−0.965 + 0.258i)12-s + (−1.21 + 1.21i)13-s + (0.101 − 2.64i)14-s + (0.500 − 0.866i)16-s + (−1.95 + 7.31i)17-s + (−0.258 + 0.965i)18-s + (−2.32 + 4.02i)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 + (−0.955 − 0.295i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.395 − 0.685i)11-s + (−0.278 + 0.0747i)12-s + (−0.337 + 0.337i)13-s + (0.0270 − 0.706i)14-s + (0.125 − 0.216i)16-s + (−0.475 + 1.77i)17-s + (−0.0610 + 0.227i)18-s + (−0.533 + 0.924i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7687452502\)
\(L(\frac12)\) \(\approx\) \(0.7687452502\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 + 0.781i)T \)
good11 \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.21 - 1.21i)T - 13iT^{2} \)
17 \( 1 + (1.95 - 7.31i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.32 - 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.95 - 1.32i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.99iT - 29T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.02 + 3.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (-0.545 - 0.545i)T + 43iT^{2} \)
47 \( 1 + (-6.12 + 1.64i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.22 + 8.28i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.86 + 6.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 2.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 + 0.663i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + (-13.1 - 3.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.78 - 4.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.99 + 7.99i)T - 83iT^{2} \)
89 \( 1 + (0.0812 - 0.140i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.35 - 4.35i)T + 97iT^{2} \)
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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

−10.35988504125457356610443494093, −9.287389644194625558628113288366, −8.639472113611547855445735965631, −7.88580910253977407554982303683, −6.96188322223746868518452904730, −6.16867076170527090262036306639, −5.35641935573627972573302605756, −3.88564800202627719734752307311, −3.59142293300102641406970418703, −2.00998384621524033289570325489, 0.27878805812518909116052307650, 2.30564646735008526844970909663, 2.76414312045829987727362706418, 4.02020401599819001298619720743, 4.91580661648568834223908850291, 6.03498839980518054232493545147, 7.05731460414519499867985149705, 7.81439961160535780005843847404, 9.023141321021712916887379912702, 9.489555121585873672894802251548

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