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Properties

Label 2-2100-105.59-c1-0-27
Degree $2$
Conductor $2100$
Sign $0.753 - 0.657i$
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  + (0.576 + 1.63i)3-s + (1.99 − 1.73i)7-s + (−2.33 + 1.88i)9-s + (3.38 + 1.95i)11-s + 6.06·13-s + (−2.65 − 1.53i)17-s + (2.94 − 1.70i)19-s + (3.98 + 2.26i)21-s + (1.43 + 2.48i)23-s + (−4.42 − 2.72i)27-s − 7.97i·29-s + (−5.63 − 3.25i)31-s + (−1.23 + 6.64i)33-s + (0.113 − 0.0654i)37-s + (3.49 + 9.90i)39-s + ⋯
L(s)  = 1  + (0.333 + 0.942i)3-s + (0.755 − 0.655i)7-s + (−0.778 + 0.628i)9-s + (1.01 + 0.588i)11-s + 1.68·13-s + (−0.643 − 0.371i)17-s + (0.676 − 0.390i)19-s + (0.869 + 0.493i)21-s + (0.299 + 0.518i)23-s + (−0.851 − 0.524i)27-s − 1.48i·29-s + (−1.01 − 0.583i)31-s + (−0.215 + 1.15i)33-s + (0.0186 − 0.0107i)37-s + (0.560 + 1.58i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.753 - 0.657i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.465813282\)
\(L(\frac12)\) \(\approx\) \(2.465813282\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.576 - 1.63i)T \)
5 \( 1 \)
7 \( 1 + (-1.99 + 1.73i)T \)
good11 \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 + (2.65 + 1.53i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.94 + 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.43 - 2.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 + 3.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.113 + 0.0654i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 4.43iT - 43T^{2} \)
47 \( 1 + (8.71 - 5.02i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.67 - 4.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.28 + 2.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.44 + 4.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.8 - 7.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + (3.88 - 6.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.22 - 2.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (-4.11 - 7.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.74T + 97T^{2} \)
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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

−9.373380889789143644391406856713, −8.483080814934777215971942536424, −7.80763907108157766101742670697, −6.90259427094008705358367635275, −5.93465033243832714730913420778, −5.03398255015271665560643186568, −4.02896051630168612301427683141, −3.82790818924466917101120604222, −2.38772060212472879885389252800, −1.13335921929443968133514638475, 1.13656032178728161454936748490, 1.81782848631241835852772147930, 3.13714435189477986695551531148, 3.86953565103044601625713861993, 5.19661952661631577751212124029, 6.07144434449125901598995727462, 6.55820347602625272802399075098, 7.52281306680557925994093077351, 8.504530695839415764486880980438, 8.682207065630492666131632206750

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