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Properties

Label 2-2100-21.17-c1-0-48
Degree $2$
Conductor $2100$
Sign $-0.463 - 0.886i$
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  + (−1.64 − 0.528i)3-s + (−1.63 − 2.08i)7-s + (2.44 + 1.74i)9-s + (−5.09 − 2.93i)11-s − 3.54i·13-s + (1.55 − 2.69i)17-s + (−3.58 + 2.06i)19-s + (1.59 + 4.29i)21-s + (6.19 − 3.57i)23-s + (−3.10 − 4.16i)27-s + 6.84i·29-s + (−3.90 − 2.25i)31-s + (6.84 + 7.53i)33-s + (0.986 + 1.70i)37-s + (−1.86 + 5.83i)39-s + ⋯
L(s)  = 1  + (−0.952 − 0.304i)3-s + (−0.617 − 0.786i)7-s + (0.814 + 0.580i)9-s + (−1.53 − 0.886i)11-s − 0.981i·13-s + (0.376 − 0.652i)17-s + (−0.822 + 0.474i)19-s + (0.347 + 0.937i)21-s + (1.29 − 0.745i)23-s + (−0.598 − 0.801i)27-s + 1.27i·29-s + (−0.701 − 0.404i)31-s + (1.19 + 1.31i)33-s + (0.162 + 0.281i)37-s + (−0.299 + 0.935i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.463 - 0.886i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.463 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05875699641\)
\(L(\frac12)\) \(\approx\) \(0.05875699641\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.528i)T \)
5 \( 1 \)
7 \( 1 + (1.63 + 2.08i)T \)
good11 \( 1 + (5.09 + 2.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.54iT - 13T^{2} \)
17 \( 1 + (-1.55 + 2.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.58 - 2.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.19 + 3.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.84iT - 29T^{2} \)
31 \( 1 + (3.90 + 2.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.986 - 1.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.33T + 41T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 + (-0.916 - 1.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.9 + 6.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.702 + 0.405i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.10 - 8.84i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.18iT - 71T^{2} \)
73 \( 1 + (-1.87 - 1.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.05 - 8.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 + (6.28 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.50iT - 97T^{2} \)
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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

−8.361250706252421544137697821501, −7.70576979859757602579785873068, −6.99760789203190322900296285211, −6.18487644702805311074513098617, −5.38719732448677087513650627977, −4.80132283272742640373227456951, −3.50694544971357630250191241476, −2.65406001065542718943171315058, −0.972133493491917054808534176028, −0.02833179865166797684511515578, 1.82524867999403389774352067033, 2.88976843823318717698532910345, 4.13020582756546077933278329432, 4.93286683104522874507169359311, 5.62852549130561813389812687665, 6.40133212809996802323318638110, 7.14965213326355789762010257556, 7.988435694692918034565937840574, 9.186030249396098387394264143411, 9.534731788758571892922133030196

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