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Properties

Label 2-2100-105.59-c1-0-38
Degree $2$
Conductor $2100$
Sign $-0.00342 + 0.999i$
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 + 6.92·13-s + (3 − 1.73i)19-s + (−1.50 − 4.33i)21-s − 5.19·27-s + (1.5 + 0.866i)31-s + (−0.866 + 0.5i)37-s + (5.99 − 10.3i)39-s + 13i·43-s + (−1.00 − 6.92i)49-s − 6i·57-s + (7.5 − 4.33i)61-s + (−7.79 − 1.5i)63-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.654 − 0.755i)7-s + (−0.5 − 0.866i)9-s + 1.92·13-s + (0.688 − 0.397i)19-s + (−0.327 − 0.944i)21-s − 1.00·27-s + (0.269 + 0.155i)31-s + (−0.142 + 0.0821i)37-s + (0.960 − 1.66i)39-s + 1.98i·43-s + (−0.142 − 0.989i)49-s − 0.794i·57-s + (0.960 − 0.554i)61-s + (−0.981 − 0.188i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00342 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00342 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.00342 + 0.999i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.00342 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.468491243\)
\(L(\frac12)\) \(\approx\) \(2.468491243\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.92T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-7.79 + 13.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−8.731025991216353705822926485441, −8.066091458369955146860783817229, −7.50845129625152301202234044478, −6.58409751788699348867760008366, −6.00528489527983275011596334860, −4.83467482829402363980460893798, −3.79431468724413045246406853962, −3.05313162314046924862413039865, −1.67507433997354592061625018928, −0.933942215196697982230963762485, 1.45226184385446884666863072064, 2.60786955615935354365664428633, 3.60791787449942870926618216281, 4.27727267234888711153582976977, 5.44883768309404275147621070106, 5.77886012682639433547324761703, 7.03558814296077391915189996431, 8.155965293505696062490457479373, 8.525259955236844505110993984269, 9.145829726633285185066940285632

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