Properties

Label 2-2100-105.59-c1-0-34
Degree $2$
Conductor $2100$
Sign $0.992 + 0.123i$
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  + 1.73·3-s + (1.73 − 2i)7-s + 2.99·9-s + (4.5 + 2.59i)11-s + (2.59 + 1.5i)17-s + (−1.5 + 0.866i)19-s + (2.99 − 3.46i)21-s + (−2.59 − 4.5i)23-s + 5.19·27-s + (−1.5 − 0.866i)31-s + (7.79 + 4.5i)33-s + (−6.06 + 3.5i)37-s + 6·41-s + 4i·43-s + (−2.59 + 1.5i)47-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.654 − 0.755i)7-s + 0.999·9-s + (1.35 + 0.783i)11-s + (0.630 + 0.363i)17-s + (−0.344 + 0.198i)19-s + (0.654 − 0.755i)21-s + (−0.541 − 0.938i)23-s + 1.00·27-s + (−0.269 − 0.155i)31-s + (1.35 + 0.783i)33-s + (−0.996 + 0.575i)37-s + 0.937·41-s + 0.609i·43-s + (−0.378 + 0.218i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.111949043\)
\(L(\frac12)\) \(\approx\) \(3.111949043\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2.59 - 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 4.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.59 + 4.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-6.06 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92T + 97T^{2} \)
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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

−8.977540157762406395696914456462, −8.348488756940266449607824689052, −7.57891493484544601645335788004, −6.96185828888425626033432062650, −6.11590173141358113435337909537, −4.67949374212584410767829872163, −4.17849430268801223419486782381, −3.37294636723242062408804669758, −2.04581514374558732256405605944, −1.26257477679841488074544912869, 1.28295683676576889501872977007, 2.21206724306692298601813684845, 3.33589164147039848152608507416, 3.99285191515913570089755753887, 5.08734011940768840149435933541, 5.97195491624406278477028424106, 6.88659130775703887294448424387, 7.77363497819103647830616856825, 8.390500013846390585104252692873, 9.187897803976598671407402709190

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