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Properties

Label 2-2100-105.89-c1-0-29
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: $\chi_{2100} (1349, \cdot )$
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

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

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