Properties

Label 2-1050-5.2-c2-0-14
Degree $2$
Conductor $1050$
Sign $0.130 - 0.991i$
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 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + 2.44·6-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s − 3.02·11-s + (2.44 + 2.44i)12-s + (−2.99 + 2.99i)13-s − 3.74i·14-s − 4·16-s + (13.0 + 13.0i)17-s + (2.99 − 2.99i)18-s + 29.5i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + 0.408·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s − 0.274·11-s + (0.204 + 0.204i)12-s + (−0.230 + 0.230i)13-s − 0.267i·14-s − 0.250·16-s + (0.768 + 0.768i)17-s + (0.166 − 0.166i)18-s + 1.55i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.527823209\)
\(L(\frac12)\) \(\approx\) \(2.527823209\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 + 3.02T + 121T^{2} \)
13 \( 1 + (2.99 - 2.99i)T - 169iT^{2} \)
17 \( 1 + (-13.0 - 13.0i)T + 289iT^{2} \)
19 \( 1 - 29.5iT - 361T^{2} \)
23 \( 1 + (-2.34 + 2.34i)T - 529iT^{2} \)
29 \( 1 - 24.4iT - 841T^{2} \)
31 \( 1 - 46.7T + 961T^{2} \)
37 \( 1 + (-30.6 - 30.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 11.9T + 1.68e3T^{2} \)
43 \( 1 + (0.402 - 0.402i)T - 1.84e3iT^{2} \)
47 \( 1 + (-46.4 - 46.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-11.9 + 11.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 32.2iT - 3.48e3T^{2} \)
61 \( 1 + 78.3T + 3.72e3T^{2} \)
67 \( 1 + (17.3 + 17.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 43.2T + 5.04e3T^{2} \)
73 \( 1 + (-39.4 + 39.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 0.100iT - 6.24e3T^{2} \)
83 \( 1 + (15.9 - 15.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 91.4iT - 7.92e3T^{2} \)
97 \( 1 + (-14.1 - 14.1i)T + 9.40e3iT^{2} \)
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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

−9.917574771202498151000632949691, −8.875906398044594675124585382247, −7.981500135098552937073994725874, −7.52986072584436940808638501180, −6.42100218552709864627322944358, −5.85846373818538058473605598537, −4.64679385085526508410327395258, −3.68735295651884381779168578991, −2.76408212666854383741654534618, −1.33277433535818801258278292828, 0.66204799252047195649048382416, 2.44694951846376066981225443970, 3.00049502515479522384554336814, 4.21375900649291110420612283657, 5.01858553335785754874960230877, 5.88060713725041557088075108700, 7.00209415806933682238573030494, 7.908065324251235663535786492685, 8.965658782885510656854827948861, 9.613792657030627081405934968838

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