Properties

Label 2-1050-35.34-c2-0-7
Degree $2$
Conductor $1050$
Sign $-0.990 - 0.137i$
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 + 1.73·3-s − 2.00·4-s + 2.44i·6-s + (6.63 − 2.24i)7-s − 2.82i·8-s + 2.99·9-s − 10.2·11-s − 3.46·12-s − 8.95·13-s + (3.17 + 9.37i)14-s + 4.00·16-s − 30.4·17-s + 4.24i·18-s + 16.1i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.500·4-s + 0.408i·6-s + (0.947 − 0.320i)7-s − 0.353i·8-s + 0.333·9-s − 0.931·11-s − 0.288·12-s − 0.689·13-s + (0.226 + 0.669i)14-s + 0.250·16-s − 1.78·17-s + 0.235i·18-s + 0.849i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.990 - 0.137i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.990 - 0.137i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.043805081\)
\(L(\frac12)\) \(\approx\) \(1.043805081\)
\(L(2)\) 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 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-6.63 + 2.24i)T \)
good11 \( 1 + 10.2T + 121T^{2} \)
13 \( 1 + 8.95T + 169T^{2} \)
17 \( 1 + 30.4T + 289T^{2} \)
19 \( 1 - 16.1iT - 361T^{2} \)
23 \( 1 - 6.72iT - 529T^{2} \)
29 \( 1 + 30T + 841T^{2} \)
31 \( 1 - 50.1iT - 961T^{2} \)
37 \( 1 - 30.9iT - 1.36e3T^{2} \)
41 \( 1 + 7.10iT - 1.68e3T^{2} \)
43 \( 1 - 74.4iT - 1.84e3T^{2} \)
47 \( 1 - 58.2T + 2.20e3T^{2} \)
53 \( 1 - 70.9iT - 2.80e3T^{2} \)
59 \( 1 - 0.492iT - 3.48e3T^{2} \)
61 \( 1 - 2.86iT - 3.72e3T^{2} \)
67 \( 1 - 27.0iT - 4.48e3T^{2} \)
71 \( 1 - 50.6T + 5.04e3T^{2} \)
73 \( 1 + 70.6T + 5.32e3T^{2} \)
79 \( 1 + 133.T + 6.24e3T^{2} \)
83 \( 1 + 104.T + 6.88e3T^{2} \)
89 \( 1 + 144. iT - 7.92e3T^{2} \)
97 \( 1 - 100.T + 9.40e3T^{2} \)
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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

−10.04669891549042893098982061721, −8.998533814762311998893049708625, −8.396865976976764231431741464448, −7.58148761798671287165562491013, −7.05948649386764507853770894561, −5.84199917894686106142460832005, −4.84817898372594019968804889881, −4.22977052652093758561460144039, −2.83306561183549535993028376963, −1.62143940326523710664262219940, 0.27618175968791770979463784608, 2.17929225163690731501125354646, 2.41792520358083221819932736173, 3.97672078498655144393589980261, 4.75393712074217288594007587042, 5.60666837318099908176916493970, 7.06466288989372708529775328900, 7.80661989931934904394265859584, 8.717555662282132615521546242908, 9.204104270990287157043386414659

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