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Properties

Label 2-1050-15.14-c2-0-24
Degree $2$
Conductor $1050$
Sign $-0.424 - 0.905i$
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  + 1.41·2-s + (2.35 + 1.86i)3-s + 2.00·4-s + (3.32 + 2.63i)6-s + 2.64i·7-s + 2.82·8-s + (2.07 + 8.75i)9-s + 19.0i·11-s + (4.70 + 3.72i)12-s + 1.55i·13-s + 3.74i·14-s + 4.00·16-s − 28.9·17-s + (2.93 + 12.3i)18-s − 23.3·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.784 + 0.620i)3-s + 0.500·4-s + (0.554 + 0.438i)6-s + 0.377i·7-s + 0.353·8-s + (0.230 + 0.973i)9-s + 1.73i·11-s + (0.392 + 0.310i)12-s + 0.119i·13-s + 0.267i·14-s + 0.250·16-s − 1.70·17-s + (0.162 + 0.688i)18-s − 1.22·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.424 - 0.905i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.408518988\)
\(L(\frac12)\) \(\approx\) \(3.408518988\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-2.35 - 1.86i)T \)
5 \( 1 \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 19.0iT - 121T^{2} \)
13 \( 1 - 1.55iT - 169T^{2} \)
17 \( 1 + 28.9T + 289T^{2} \)
19 \( 1 + 23.3T + 361T^{2} \)
23 \( 1 - 23.7T + 529T^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 - 7.88T + 961T^{2} \)
37 \( 1 + 20.5iT - 1.36e3T^{2} \)
41 \( 1 + 50.8iT - 1.68e3T^{2} \)
43 \( 1 - 31.2iT - 1.84e3T^{2} \)
47 \( 1 - 15.7T + 2.20e3T^{2} \)
53 \( 1 - 78.7T + 2.80e3T^{2} \)
59 \( 1 - 89.1iT - 3.48e3T^{2} \)
61 \( 1 - 56.4T + 3.72e3T^{2} \)
67 \( 1 - 56.2iT - 4.48e3T^{2} \)
71 \( 1 - 28.4iT - 5.04e3T^{2} \)
73 \( 1 - 120. iT - 5.32e3T^{2} \)
79 \( 1 - 72.1T + 6.24e3T^{2} \)
83 \( 1 - 104.T + 6.88e3T^{2} \)
89 \( 1 - 36.3iT - 7.92e3T^{2} \)
97 \( 1 + 57.1iT - 9.40e3T^{2} \)
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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

−10.01334752502416946058487394563, −9.100477341878331998310690574359, −8.509969187448757135580894001649, −7.30298928620244160950301781131, −6.74282600335624263051722135989, −5.43552703253037991229989367012, −4.42869731381429788772312686259, −4.12099313915714542658084534394, −2.53935293842368759740078041605, −2.07669200810697804722316756641, 0.69564180520435112141336066383, 2.13342220849169375649651527579, 3.12406784676141409117859255372, 3.93174447989328491721841260091, 5.02874132349377768794627447465, 6.42155771208699834418802354781, 6.59144136416269180304387794357, 7.81591279604723675408298171630, 8.595708115265162844255676965614, 9.129371628221410481575199339883

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