Properties

Label 2-1050-3.2-c2-0-5
Degree $2$
Conductor $1050$
Sign $-0.569 - 0.821i$
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 + (−2.46 + 1.70i)3-s − 2.00·4-s + (2.41 + 3.48i)6-s − 2.64·7-s + 2.82i·8-s + (3.15 − 8.42i)9-s − 2.66i·11-s + (4.93 − 3.41i)12-s + 5.56·13-s + 3.74i·14-s + 4.00·16-s + 18.5i·17-s + (−11.9 − 4.46i)18-s − 0.634·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.821 + 0.569i)3-s − 0.500·4-s + (0.402 + 0.581i)6-s − 0.377·7-s + 0.353i·8-s + (0.350 − 0.936i)9-s − 0.242i·11-s + (0.410 − 0.284i)12-s + 0.427·13-s + 0.267i·14-s + 0.250·16-s + 1.09i·17-s + (−0.662 − 0.248i)18-s − 0.0334·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.569 - 0.821i$
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: \((2,\ 1050,\ (\ :1),\ -0.569 - 0.821i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2567289965\)
\(L(\frac12)\) \(\approx\) \(0.2567289965\)
\(L(2)\) 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 \( 1 + (2.46 - 1.70i)T \)
5 \( 1 \)
7 \( 1 + 2.64T \)
good11 \( 1 + 2.66iT - 121T^{2} \)
13 \( 1 - 5.56T + 169T^{2} \)
17 \( 1 - 18.5iT - 289T^{2} \)
19 \( 1 + 0.634T + 361T^{2} \)
23 \( 1 + 15.6iT - 529T^{2} \)
29 \( 1 + 43.3iT - 841T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 + 35.0T + 1.36e3T^{2} \)
41 \( 1 + 15.6iT - 1.68e3T^{2} \)
43 \( 1 - 64.4T + 1.84e3T^{2} \)
47 \( 1 - 52.5iT - 2.20e3T^{2} \)
53 \( 1 - 51.6iT - 2.80e3T^{2} \)
59 \( 1 - 32.9iT - 3.48e3T^{2} \)
61 \( 1 + 104.T + 3.72e3T^{2} \)
67 \( 1 + 113.T + 4.48e3T^{2} \)
71 \( 1 - 36.8iT - 5.04e3T^{2} \)
73 \( 1 + 144.T + 5.32e3T^{2} \)
79 \( 1 + 57.9T + 6.24e3T^{2} \)
83 \( 1 - 45.0iT - 6.88e3T^{2} \)
89 \( 1 + 80.9iT - 7.92e3T^{2} \)
97 \( 1 + 96.2T + 9.40e3T^{2} \)
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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

−10.40901115759326911660378748911, −9.320871524082514149292749408101, −8.691154940296191124076227323143, −7.54844123170706152279719260611, −6.21578602484559248484159621317, −5.84799878901932131172382109519, −4.51904675290480177645537106559, −3.93400081845314786276320355179, −2.79333082906353530752914376356, −1.22818858274992730409531432507, 0.10073299613356455525567657452, 1.48267898968041601208883453584, 3.09684167004689861298138966689, 4.46873094901449364921569297729, 5.32598333496649503715631540948, 6.05910256735177285979191281416, 7.00836489322940191149667930697, 7.38722736374602377748253114056, 8.493753896389366252124908626851, 9.344194301402840365068768641066

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