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Properties

Label 2-2100-21.5-c1-0-23
Degree $2$
Conductor $2100$
Sign $0.484 - 0.874i$
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  + (1.72 + 0.134i)3-s + (−0.786 + 2.52i)7-s + (2.96 + 0.463i)9-s + (2.34 − 1.35i)11-s + 1.12i·13-s + (3.69 + 6.39i)17-s + (−0.412 − 0.238i)19-s + (−1.69 + 4.25i)21-s + (−4.84 − 2.79i)23-s + (5.05 + 1.19i)27-s + 2.20i·29-s + (−2.07 + 1.19i)31-s + (4.23 − 2.02i)33-s + (−4.34 + 7.53i)37-s + (−0.150 + 1.94i)39-s + ⋯
L(s)  = 1  + (0.997 + 0.0774i)3-s + (−0.297 + 0.954i)7-s + (0.988 + 0.154i)9-s + (0.706 − 0.408i)11-s + 0.311i·13-s + (0.895 + 1.55i)17-s + (−0.0945 − 0.0546i)19-s + (−0.370 + 0.928i)21-s + (−1.01 − 0.583i)23-s + (0.973 + 0.230i)27-s + 0.409i·29-s + (−0.372 + 0.214i)31-s + (0.736 − 0.352i)33-s + (−0.714 + 1.23i)37-s + (−0.0241 + 0.311i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.484 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.569540309\)
\(L(\frac12)\) \(\approx\) \(2.569540309\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.134i)T \)
5 \( 1 \)
7 \( 1 + (0.786 - 2.52i)T \)
good11 \( 1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.12iT - 13T^{2} \)
17 \( 1 + (-3.69 - 6.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.412 + 0.238i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.84 + 2.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.20iT - 29T^{2} \)
31 \( 1 + (2.07 - 1.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.34 - 7.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.42T + 41T^{2} \)
43 \( 1 + 4.16T + 43T^{2} \)
47 \( 1 + (-6.21 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.25 + 2.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.15 - 2.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 + 1.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.34 - 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.89iT - 71T^{2} \)
73 \( 1 + (6.29 - 3.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.47 + 6.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (-3.48 + 6.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.79iT - 97T^{2} \)
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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

−8.941166631765225626111123751894, −8.626471609844748704596074284053, −7.952183173321485771131442954076, −6.86953058885138905048042475409, −6.17000829040390262143133563177, −5.28709000914016793680030521773, −4.03181201409035283520046270060, −3.49209205522742079100188458997, −2.40690052897455425020110554301, −1.49045001189378951719265864930, 0.844724425353209600631603986764, 2.06804466300406547384058116458, 3.21490368772096430654837436730, 3.88322905142614884672633668011, 4.70396854856378503490311562658, 5.89458989079301467965779315097, 6.92310944852146239210001905010, 7.52339502829261209441008843432, 7.975328490803047143385240661200, 9.290237563281056143460048263921

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