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Properties

Label 2-2100-105.62-c0-0-2
Degree $2$
Conductor $2100$
Sign $0.0706 - 0.997i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.707 + 0.707i)13-s − 1.73·19-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s − 1.73i·31-s + 1.00i·39-s + (1.22 − 1.22i)43-s + (−0.866 + 0.499i)49-s + (−1.22 − 1.22i)57-s + 1.73i·61-s + (−0.965 + 0.258i)63-s + (1.22 + 1.22i)67-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.707 + 0.707i)13-s − 1.73·19-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s − 1.73i·31-s + 1.00i·39-s + (1.22 − 1.22i)43-s + (−0.866 + 0.499i)49-s + (−1.22 − 1.22i)57-s + 1.73i·61-s + (−0.965 + 0.258i)63-s + (1.22 + 1.22i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0706 - 0.997i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (2057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ 0.0706 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478263268\)
\(L(\frac12)\) \(\approx\) \(1.478263268\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-0.258 - 0.965i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−9.314367506554107893165909924810, −8.684351906742652740344929119883, −8.293475794220688867863316582286, −7.26866737789852413349185359918, −6.18661038426836517400163536251, −5.51416036628367077679939570871, −4.38684835536315833433932550365, −3.90742980170201014452337436259, −2.59860272279242176024236635335, −1.96360172655586666706142522880, 1.01380578113659531239758632207, 2.12040166946420337732750179844, 3.28834153092104704587477894295, 4.00095611108741010692426345730, 5.02761707728075642182977994399, 6.33827192082420388244126585189, 6.69596310019873133251717065592, 7.76944351024615759340716469239, 8.173690481748484119114611249295, 8.921161231531229771691319567445

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