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Properties

Label 2-2100-21.20-c1-0-30
Degree $2$
Conductor $2100$
Sign $0.952 - 0.303i$
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  + (1.67 + 0.437i)3-s + (2.23 − 1.41i)7-s + (2.61 + 1.46i)9-s + 3.83i·11-s − 5.99i·13-s + 2.07·17-s + 5.11i·19-s + (4.36 − 1.39i)21-s + 8.57i·23-s + (3.74 + 3.59i)27-s − 5.64i·29-s − 1.95i·31-s + (−1.67 + 6.42i)33-s + 2.23·37-s + (2.61 − 10.0i)39-s + ⋯
L(s)  = 1  + (0.967 + 0.252i)3-s + (0.845 − 0.534i)7-s + (0.872 + 0.488i)9-s + 1.15i·11-s − 1.66i·13-s + 0.502·17-s + 1.17i·19-s + (0.952 − 0.303i)21-s + 1.78i·23-s + (0.721 + 0.692i)27-s − 1.04i·29-s − 0.351i·31-s + (−0.291 + 1.11i)33-s + 0.367·37-s + (0.419 − 1.60i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.952 - 0.303i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.948601632\)
\(L(\frac12)\) \(\approx\) \(2.948601632\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 0.437i)T \)
5 \( 1 \)
7 \( 1 + (-2.23 + 1.41i)T \)
good11 \( 1 - 3.83iT - 11T^{2} \)
13 \( 1 + 5.99iT - 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 - 5.11iT - 19T^{2} \)
23 \( 1 - 8.57iT - 23T^{2} \)
29 \( 1 + 5.64iT - 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
37 \( 1 - 2.23T + 37T^{2} \)
41 \( 1 - 7.49T + 41T^{2} \)
43 \( 1 + 9.47T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 4.37iT - 61T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 - 1.41iT - 73T^{2} \)
79 \( 1 + 7.47T + 79T^{2} \)
83 \( 1 - 7.49T + 83T^{2} \)
89 \( 1 + 2.86T + 89T^{2} \)
97 \( 1 + 0.206iT - 97T^{2} \)
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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

−9.203287016898768924668595787820, −7.997365511211711118612161201120, −7.85600514867604608321934260805, −7.23688323747422908240273523607, −5.80603145890195078636400235106, −5.04953358471107180688863807220, −4.09950572995656270924176089441, −3.41629638727521401790751017756, −2.25066896069762716418436420688, −1.27535629785381147411643249239, 1.13173260494043778798179546691, 2.25530797026107518215572106439, 3.01083950561464785486542455860, 4.19314072315191528425152942496, 4.84254906890160343434109646763, 6.06054233844230194080531626022, 6.81891953260303186134281326141, 7.59949712784544369867777931158, 8.600622256370609477106562050208, 8.794726358498969353824264504535

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