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Properties

Label 2-1050-35.19-c2-0-22
Degree $2$
Conductor $1050$
Sign $-0.129 - 0.991i$
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  + (1.22 + 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + 2.44i·6-s + (−4.61 − 5.26i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (5.41 + 9.37i)11-s + (−1.73 + 2.99i)12-s + 19.2·13-s + (−1.93 − 9.70i)14-s + (−2.00 + 3.46i)16-s + (−5.13 − 8.89i)17-s + (−3.67 + 2.12i)18-s + (18.0 + 10.4i)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 + (−0.659 − 0.751i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.491 + 0.852i)11-s + (−0.144 + 0.249i)12-s + 1.48·13-s + (−0.137 − 0.693i)14-s + (−0.125 + 0.216i)16-s + (−0.302 − 0.523i)17-s + (−0.204 + 0.117i)18-s + (0.951 + 0.549i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.129 - 0.991i$
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: \((2,\ 1050,\ (\ :1),\ -0.129 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.987334524\)
\(L(\frac12)\) \(\approx\) \(2.987334524\)
\(L(2)\) 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 \( 1 + (4.61 + 5.26i)T \)
good11 \( 1 + (-5.41 - 9.37i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 19.2T + 169T^{2} \)
17 \( 1 + (5.13 + 8.89i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-18.0 - 10.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.2 - 10.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 19.0T + 841T^{2} \)
31 \( 1 + (34.6 - 20.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-43.6 - 25.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.7iT - 1.68e3T^{2} \)
43 \( 1 - 48.4iT - 1.84e3T^{2} \)
47 \( 1 + (33.2 - 57.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (4.28 - 2.47i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-24.4 + 14.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.6 + 35.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.7 + 9.65i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 49.4T + 5.04e3T^{2} \)
73 \( 1 + (-66.4 - 115. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (45.0 - 78.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 101.T + 6.88e3T^{2} \)
89 \( 1 + (34.3 + 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 68.6T + 9.40e3T^{2} \)
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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

−9.684710561405265611398580090237, −9.339982248466565187139138013466, −8.167659392319640695051603510166, −7.30224588264937506633545000796, −6.58301297532054787750097793045, −5.63288741972131242025597125966, −4.58118150656952848865987993425, −3.75035587884931853008244656624, −3.08525270344207983788110662111, −1.38083047487358458783833425763, 0.78617217198218185358942872412, 2.08345821703541328750839488120, 3.24379048169321835351011932821, 3.81034756158478154574836160754, 5.33535657899121151121847444645, 6.09459319503691447985754070128, 6.66729336954908964419378612898, 7.83747073520572627578864169035, 9.019196737577057923803581162411, 9.120538644482995522691102000578

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