Properties

Label 2-1050-5.3-c2-0-12
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 + 4.18·11-s + (2.44 − 2.44i)12-s + (−12.9 − 12.9i)13-s − 3.74i·14-s − 4·16-s + (23.0 − 23.0i)17-s + (−2.99 − 2.99i)18-s + 32.6i·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 + 0.380·11-s + (0.204 − 0.204i)12-s + (−0.999 − 0.999i)13-s − 0.267i·14-s − 0.250·16-s + (1.35 − 1.35i)17-s + (−0.166 − 0.166i)18-s + 1.71i·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: $\chi_{1050} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.518446305\)
\(L(\frac12)\) \(\approx\) \(1.518446305\)
\(L(2)\) 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 \( 1 - 4.18T + 121T^{2} \)
13 \( 1 + (12.9 + 12.9i)T + 169iT^{2} \)
17 \( 1 + (-23.0 + 23.0i)T - 289iT^{2} \)
19 \( 1 - 32.6iT - 361T^{2} \)
23 \( 1 + (-13.5 - 13.5i)T + 529iT^{2} \)
29 \( 1 - 23.2iT - 841T^{2} \)
31 \( 1 - 32.2T + 961T^{2} \)
37 \( 1 + (-21.0 + 21.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 51.9T + 1.68e3T^{2} \)
43 \( 1 + (-24.7 - 24.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (52.0 - 52.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (26.3 + 26.3i)T + 2.80e3iT^{2} \)
59 \( 1 - 103. iT - 3.48e3T^{2} \)
61 \( 1 - 119.T + 3.72e3T^{2} \)
67 \( 1 + (-49.4 + 49.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 94.0T + 5.04e3T^{2} \)
73 \( 1 + (-15.0 - 15.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 112. iT - 6.24e3T^{2} \)
83 \( 1 + (13.6 + 13.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 35.6iT - 7.92e3T^{2} \)
97 \( 1 + (72.1 - 72.1i)T - 9.40e3iT^{2} \)
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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

−9.823206503283132915701197248039, −9.277113336323105610228512047474, −8.110234848385367855970145984745, −7.71396575436347263927055721892, −6.71908174879233681408713450377, −5.55500196125029209464640519757, −5.03089844130532868487980361779, −3.58029450445980521398017193279, −2.68624666211677724334210024954, −1.08370642359875662801880795558, 0.62905877907923150572258735944, 1.90092172207297677750823483593, 2.91287027929317081897136838536, 3.95408415441468099800470654092, 4.97910151505228025876763398364, 6.51698267560434873313616204110, 6.97465456093044616987767608976, 8.030076963560764700602040402391, 8.659040219254798113627808446997, 9.655494491933572613215856442820

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