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Properties

Label 2-2100-105.59-c1-0-24
Degree $2$
Conductor $2100$
Sign $0.228 - 0.973i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.228 - 0.973i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.708283426\)
\(L(\frac12)\) \(\approx\) \(2.708283426\)
\(L(\frac{3}{2})\) 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

−9.295407131066777876938833333678, −8.545785836546510779290329440597, −7.84200334428009396297134731062, −7.21883425236933596403045803361, −5.83143845099523920587861752983, −5.24876520204660364283619618941, −4.33857563333586445553592820094, −3.51426868221559569715361652250, −2.51801336216886820377060501703, −1.47107352662560964979535891801, 1.01106918925864212033719364349, 1.72088291803577679447114391395, 3.23361170106630993125032900647, 3.64154780714100277043619314027, 5.02061468777161446105735395552, 5.74372510257879710141451712859, 6.87787452640623553686027211975, 7.56585079525242644082492237864, 7.905277958234286112770610154574, 8.889352283072763258241030457636

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