Properties

Label 2-2100-105.59-c1-0-41
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  + (−1.19 − 1.25i)3-s + (−2.10 − 1.60i)7-s + (−0.157 + 2.99i)9-s + (2.05 + 1.18i)11-s + 0.748·13-s + (−6.53 − 3.77i)17-s + (6.11 − 3.53i)19-s + (0.485 + 4.55i)21-s + (1.63 + 2.83i)23-s + (3.95 − 3.37i)27-s − 2.48i·29-s + (−6.84 − 3.95i)31-s + (−0.961 − 4.00i)33-s + (−3.73 + 2.15i)37-s + (−0.892 − 0.940i)39-s + ⋯
L(s)  = 1  + (−0.688 − 0.725i)3-s + (−0.794 − 0.607i)7-s + (−0.0523 + 0.998i)9-s + (0.620 + 0.358i)11-s + 0.207·13-s + (−1.58 − 0.914i)17-s + (1.40 − 0.810i)19-s + (0.105 + 0.994i)21-s + (0.340 + 0.590i)23-s + (0.760 − 0.649i)27-s − 0.461i·29-s + (−1.22 − 0.709i)31-s + (−0.167 − 0.696i)33-s + (−0.614 + 0.354i)37-s + (−0.142 − 0.150i)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: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.641 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04136018070\)
\(L(\frac12)\) \(\approx\) \(0.04136018070\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.25i)T \)
5 \( 1 \)
7 \( 1 + (2.10 + 1.60i)T \)
good11 \( 1 + (-2.05 - 1.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.748T + 13T^{2} \)
17 \( 1 + (6.53 + 3.77i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.11 + 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.63 - 2.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.48iT - 29T^{2} \)
31 \( 1 + (6.84 + 3.95i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 - 2.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 3.03iT - 43T^{2} \)
47 \( 1 + (5.59 - 3.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0540 - 0.0935i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.60 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 + 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 2.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-0.780 + 1.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.27 - 2.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.901iT - 83T^{2} \)
89 \( 1 + (2.43 + 4.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.662906103580786915599133672083, −7.42386390799779425322970961641, −7.03426329259313815542939356438, −6.48375392346495612945575318265, −5.45303774549590606308559029448, −4.68152490707685760366921850093, −3.61804289579039500817370119208, −2.48898721974279447351735686171, −1.24139756287219517486092700061, −0.01701547615233656921333713415, 1.68042906585622859968023692923, 3.27026749972838462038020945292, 3.74266336886693767374131882314, 4.91722387468802367810185962165, 5.61276481735337669826953299395, 6.48007941584084100214031153881, 6.85746639530047909867103666175, 8.336801855373012174408084005814, 8.986638245894217401248394225474, 9.531377458386954452061259922959

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