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Properties

Label 2-1050-35.24-c2-0-18
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: $\chi_{1050} (199, \cdot )$
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

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

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