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Properties

Label 2-2100-21.17-c1-0-44
Degree $2$
Conductor $2100$
Sign $-0.641 + 0.767i$
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  + (0.747 + 1.56i)3-s + (−0.786 − 2.52i)7-s + (−1.88 + 2.33i)9-s + (−2.34 − 1.35i)11-s − 1.12i·13-s + (−3.69 + 6.39i)17-s + (−0.412 + 0.238i)19-s + (3.35 − 3.11i)21-s + (4.84 − 2.79i)23-s + (−5.05 − 1.19i)27-s + 2.20i·29-s + (−2.07 − 1.19i)31-s + (0.362 − 4.67i)33-s + (−4.34 − 7.53i)37-s + (1.75 − 0.840i)39-s + ⋯
L(s)  = 1  + (0.431 + 0.902i)3-s + (−0.297 − 0.954i)7-s + (−0.627 + 0.778i)9-s + (−0.706 − 0.408i)11-s − 0.311i·13-s + (−0.895 + 1.55i)17-s + (−0.0945 + 0.0546i)19-s + (0.732 − 0.680i)21-s + (1.01 − 0.583i)23-s + (−0.973 − 0.230i)27-s + 0.409i·29-s + (−0.372 − 0.214i)31-s + (0.0631 − 0.813i)33-s + (−0.714 − 1.23i)37-s + (0.281 − 0.134i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.641 + 0.767i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.641 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2669374384\)
\(L(\frac12)\) \(\approx\) \(0.2669374384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.747 - 1.56i)T \)
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

−8.664677164356353957577937590552, −8.341867046748530299776969121401, −7.33519253409715654257983930682, −6.49258204508807927428182083163, −5.47596001042444651527341843584, −4.67939254605215648085406582970, −3.78754844877239642502532750243, −3.17887142375821612802121507276, −1.93329794779777052691863968850, −0.081430459580887542773749383370, 1.61093622894084109361367360472, 2.63590616066543124523520505750, 3.17275887516661584140000076502, 4.72903664904424043083340336215, 5.41230613461002540362202176511, 6.50261619521020000434157595656, 6.98220864905132867036459197938, 7.83135221701645492921817594563, 8.621877567351701021662394075453, 9.264232949347153772402512393639

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