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Properties

Label 2-1050-35.19-c2-0-28
Degree $2$
Conductor $1050$
Sign $0.286 + 0.958i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s + 2.44i·6-s + (0.325 − 6.99i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−6.09 − 10.5i)11-s + (1.73 − 2.99i)12-s + 25.3·13-s + (−5.34 + 8.33i)14-s + (−2.00 + 3.46i)16-s + (14.3 + 24.9i)17-s + (3.67 − 2.12i)18-s + (13.9 + 8.03i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + 0.408i·6-s + (0.0465 − 0.998i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.554 − 0.959i)11-s + (0.144 − 0.249i)12-s + 1.95·13-s + (−0.381 + 0.595i)14-s + (−0.125 + 0.216i)16-s + (0.846 + 1.46i)17-s + (0.204 − 0.117i)18-s + (0.732 + 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.418514330\)
\(L(\frac12)\) \(\approx\) \(1.418514330\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.325 + 6.99i)T \)
good11 \( 1 + (6.09 + 10.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 25.3T + 169T^{2} \)
17 \( 1 + (-14.3 - 24.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.9 - 8.03i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.5 - 11.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 27.9T + 841T^{2} \)
31 \( 1 + (-20.0 + 11.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (25.1 + 14.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 56.9iT - 1.68e3T^{2} \)
43 \( 1 - 7.83iT - 1.84e3T^{2} \)
47 \( 1 + (11.3 - 19.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-42.0 + 24.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-62.3 + 36.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-99.2 - 57.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-61.0 + 35.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.41T + 5.04e3T^{2} \)
73 \( 1 + (19.7 + 34.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (27.3 - 47.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 135.T + 6.88e3T^{2} \)
89 \( 1 + (124. + 72.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 78.9T + 9.40e3T^{2} \)
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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

−9.693437795767661949994422203315, −8.388815460151891646814258082421, −8.166700028046625079382342865413, −7.17763305860675184525406737178, −6.20341278221444742186191448527, −5.50668804405706326832031372528, −3.84259865717023704004721290667, −3.28948987280456550231630564335, −1.51326250039522907618376614901, −0.831753234579385434487205820716, 0.925387670476437200553734252637, 2.43554110453934168747457539936, 3.58652290567527176679206490225, 5.12256581455065498793914206037, 5.43102517727385770926345131168, 6.58609208415514546179047986337, 7.37571208784638688270273069356, 8.471956793129477632005642186573, 9.026109636677107856746339881381, 9.775387070764430194715018781499

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