Properties

Label 2-2100-105.104-c1-0-24
Degree $2$
Conductor $2100$
Sign $-0.608 + 0.793i$
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  + (−0.866 − 1.5i)3-s + (−1.73 + 2i)7-s + (−1.5 + 2.59i)9-s + 5.19i·11-s − 1.73·13-s − 3i·17-s + 3.46i·19-s + (4.5 + 0.866i)21-s + 5.19·27-s − 5.19i·29-s − 10.3i·31-s + (7.79 − 4.5i)33-s − 8i·37-s + (1.49 + 2.59i)39-s − 6·41-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.654 + 0.755i)7-s + (−0.5 + 0.866i)9-s + 1.56i·11-s − 0.480·13-s − 0.727i·17-s + 0.794i·19-s + (0.981 + 0.188i)21-s + 1.00·27-s − 0.964i·29-s − 1.86i·31-s + (1.35 − 0.783i)33-s − 1.31i·37-s + (0.240 + 0.416i)39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.608 + 0.793i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5554133086\)
\(L(\frac12)\) \(\approx\) \(0.5554133086\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (1.73 - 2i)T \)
good11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 5.19iT - 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 1.73T + 97T^{2} \)
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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

−8.831510459103064681686526486926, −7.79120176218317571177365844563, −7.28333527499410998120756304680, −6.51693599434094307351684726919, −5.71440951791959891316779765820, −5.02625452471710831901363883550, −3.93703435485964810682027349162, −2.48036418857456482128091939207, −1.98847234000340149197256220325, −0.23651482936252025973018266719, 1.06698251395381433681738102266, 3.14986421978535054933454162894, 3.41629760334448815222524742358, 4.63569651190384795753380172581, 5.25910953084939397496965856388, 6.42093398877440082922282087012, 6.63155936229976251926021155961, 7.979899222562077170325040595137, 8.725312135373934393243909695588, 9.459765999918750534224674875178

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