Properties

Label 2-2100-105.104-c1-0-14
Degree $2$
Conductor $2100$
Sign $0.988 - 0.154i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.437 − 1.67i)3-s + (−1.41 + 2.23i)7-s + (−2.61 + 1.46i)9-s − 3.83i·11-s + 5.99·13-s + 2.07i·17-s + 5.11i·19-s + (4.36 + 1.39i)21-s − 8.57·23-s + (3.59 + 3.74i)27-s − 5.64i·29-s + 1.95i·31-s + (−6.42 + 1.67i)33-s + 2.23i·37-s + (−2.61 − 10.0i)39-s + ⋯
L(s)  = 1  + (−0.252 − 0.967i)3-s + (−0.534 + 0.845i)7-s + (−0.872 + 0.488i)9-s − 1.15i·11-s + 1.66·13-s + 0.502i·17-s + 1.17i·19-s + (0.952 + 0.303i)21-s − 1.78·23-s + (0.692 + 0.721i)27-s − 1.04i·29-s + 0.351i·31-s + (−1.11 + 0.291i)33-s + 0.367i·37-s + (−0.419 − 1.60i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.988 - 0.154i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.988 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335272825\)
\(L(\frac12)\) \(\approx\) \(1.335272825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.437 + 1.67i)T \)
5 \( 1 \)
7 \( 1 + (1.41 - 2.23i)T \)
good11 \( 1 + 3.83iT - 11T^{2} \)
13 \( 1 - 5.99T + 13T^{2} \)
17 \( 1 - 2.07iT - 17T^{2} \)
19 \( 1 - 5.11iT - 19T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
29 \( 1 + 5.64iT - 29T^{2} \)
31 \( 1 - 1.95iT - 31T^{2} \)
37 \( 1 - 2.23iT - 37T^{2} \)
41 \( 1 - 7.49T + 41T^{2} \)
43 \( 1 - 9.47iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 4.37iT - 61T^{2} \)
67 \( 1 - 6.70iT - 67T^{2} \)
71 \( 1 - 2.02iT - 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 7.47T + 79T^{2} \)
83 \( 1 + 7.49iT - 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 + 0.206T + 97T^{2} \)
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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

−8.832302499103250267920024179170, −8.266378089380003230012071977492, −7.81795396449256128834390618776, −6.38992956713413019946099393959, −6.01440273631424871935223480142, −5.71273212517133650110763110304, −4.04915060081121747286031333242, −3.20311844692140993030859700806, −2.12428122809745659505676058358, −1.00295794850219311341446362886, 0.60459466585064258833363547519, 2.28163792378537708160198213287, 3.67121232143915659758899248170, 3.99065566080909261731338994945, 4.98483000236241725771549501717, 5.87452388055446410228848955606, 6.71981161184837050457021442841, 7.40623648392774153711543669685, 8.569346552414660028810585605389, 9.134022656257369220223499324978

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