Properties

Label 2-1050-3.2-c2-0-64
Degree $2$
Conductor $1050$
Sign $-0.982 + 0.184i$
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  − 1.41i·2-s + (0.554 + 2.94i)3-s − 2.00·4-s + (4.16 − 0.783i)6-s − 2.64·7-s + 2.82i·8-s + (−8.38 + 3.26i)9-s + 2.58i·11-s + (−1.10 − 5.89i)12-s + 0.180·13-s + 3.74i·14-s + 4.00·16-s − 7.25i·17-s + (4.62 + 11.8i)18-s + 21.4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.184 + 0.982i)3-s − 0.500·4-s + (0.694 − 0.130i)6-s − 0.377·7-s + 0.353i·8-s + (−0.931 + 0.363i)9-s + 0.234i·11-s + (−0.0923 − 0.491i)12-s + 0.0138·13-s + 0.267i·14-s + 0.250·16-s − 0.426i·17-s + (0.256 + 0.658i)18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.982 + 0.184i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.982 + 0.184i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.03216080382\)
\(L(\frac12)\) \(\approx\) \(0.03216080382\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-0.554 - 2.94i)T \)
5 \( 1 \)
7 \( 1 + 2.64T \)
good11 \( 1 - 2.58iT - 121T^{2} \)
13 \( 1 - 0.180T + 169T^{2} \)
17 \( 1 + 7.25iT - 289T^{2} \)
19 \( 1 - 21.4T + 361T^{2} \)
23 \( 1 - 11.8iT - 529T^{2} \)
29 \( 1 - 29.5iT - 841T^{2} \)
31 \( 1 + 49.5T + 961T^{2} \)
37 \( 1 + 43.2T + 1.36e3T^{2} \)
41 \( 1 + 20.7iT - 1.68e3T^{2} \)
43 \( 1 + 74.2T + 1.84e3T^{2} \)
47 \( 1 + 7.05iT - 2.20e3T^{2} \)
53 \( 1 + 55.0iT - 2.80e3T^{2} \)
59 \( 1 + 100. iT - 3.48e3T^{2} \)
61 \( 1 - 8.44T + 3.72e3T^{2} \)
67 \( 1 + 85.7T + 4.48e3T^{2} \)
71 \( 1 + 97.8iT - 5.04e3T^{2} \)
73 \( 1 + 17.8T + 5.32e3T^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 + 3.08iT - 6.88e3T^{2} \)
89 \( 1 + 80.1iT - 7.92e3T^{2} \)
97 \( 1 + 105.T + 9.40e3T^{2} \)
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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

−9.421692978163260796921402868534, −8.920141166664153362326613727692, −7.86743911103664676878172149084, −6.83804391312518090428334411317, −5.42454749149776538327975992054, −4.97535473598687636511456381024, −3.63921765150960248806510056251, −3.21483379347415430511665410851, −1.82639560899904908048630173898, −0.009684149920124463942203587922, 1.40334898769794615082367206570, 2.84239738226892349530147843202, 3.86867641400019364489840055305, 5.29043451455218407659060904579, 5.99899660761363210499455364289, 6.84903350445627998636440393519, 7.50105954227122897524318398444, 8.318017362601937186720193354374, 9.026424533113736664500426231363, 9.879772974776626206624000183499

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