Properties

Label 2-2100-21.20-c1-0-3
Degree $2$
Conductor $2100$
Sign $-0.0515 - 0.998i$
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 − 1.41i)3-s + (2.23 + 1.41i)7-s + (−1.00 + 2.82i)9-s − 2.82i·11-s + 2.82i·13-s − 4·17-s + 6.32i·19-s + (−0.236 − 4.57i)21-s − 6.32i·23-s + (5.00 − 1.41i)27-s + 5.65i·29-s + 6.32i·31-s + (−4.00 + 2.82i)33-s − 8.94·37-s + (4.00 − 2.82i)39-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + (0.845 + 0.534i)7-s + (−0.333 + 0.942i)9-s − 0.852i·11-s + 0.784i·13-s − 0.970·17-s + 1.45i·19-s + (−0.0515 − 0.998i)21-s − 1.31i·23-s + (0.962 − 0.272i)27-s + 1.05i·29-s + 1.13i·31-s + (−0.696 + 0.492i)33-s − 1.47·37-s + (0.640 − 0.452i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0515 - 0.998i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0515 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7621418722\)
\(L(\frac12)\) \(\approx\) \(0.7621418722\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.41i)T \)
5 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 + 6.32iT - 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 + 8.94T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 6.32iT - 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4.47T + 67T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−8.909367264878164647707441542092, −8.536677199761195456131364933167, −7.83208104239780186664027352006, −6.73184966724650541116701084692, −6.35367505677884087950366373125, −5.29537363513097536431220877157, −4.76552109722015745833079237049, −3.45110875732400806807329443058, −2.15627677858320513954708849280, −1.41748173979352115230614957211, 0.28731994301051079302390990365, 1.83415436008972393612715444535, 3.17486796212394515135986553542, 4.24726841395281649319856419949, 4.79951672695134963801350477065, 5.49810592099559209589744924158, 6.55570441124959709725569137623, 7.31266162343814268953820845971, 8.130754989241023510821439854685, 9.087690400898391079479064607155

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