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Properties

Label 2-2100-21.17-c1-0-15
Degree $2$
Conductor $2100$
Sign $0.992 + 0.122i$
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  + (−0.417 − 1.68i)3-s + (1.25 − 2.33i)7-s + (−2.65 + 1.40i)9-s + (4.63 + 2.67i)11-s + 4.13i·13-s + (0.0773 − 0.134i)17-s + (−3.40 + 1.96i)19-s + (−4.44 − 1.13i)21-s + (5.19 − 2.99i)23-s + (3.46 + 3.86i)27-s + 10.3i·29-s + (6.70 + 3.86i)31-s + (2.56 − 8.91i)33-s + (5.24 + 9.07i)37-s + (6.95 − 1.72i)39-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)3-s + (0.473 − 0.880i)7-s + (−0.883 + 0.468i)9-s + (1.39 + 0.807i)11-s + 1.14i·13-s + (0.0187 − 0.0325i)17-s + (−0.781 + 0.450i)19-s + (−0.968 − 0.247i)21-s + (1.08 − 0.625i)23-s + (0.667 + 0.744i)27-s + 1.91i·29-s + (1.20 + 0.694i)31-s + (0.446 − 1.55i)33-s + (0.861 + 1.49i)37-s + (1.11 − 0.276i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.992 + 0.122i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736223490\)
\(L(\frac12)\) \(\approx\) \(1.736223490\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.417 + 1.68i)T \)
5 \( 1 \)
7 \( 1 + (-1.25 + 2.33i)T \)
good11 \( 1 + (-4.63 - 2.67i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.13iT - 13T^{2} \)
17 \( 1 + (-0.0773 + 0.134i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.40 - 1.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 2.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + (-6.70 - 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.33T + 41T^{2} \)
43 \( 1 + 1.78T + 43T^{2} \)
47 \( 1 + (1.80 + 3.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.29 + 2.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.27 - 9.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.25 - 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.444 - 0.770i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-10.6 - 6.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.61 - 6.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.14T + 83T^{2} \)
89 \( 1 + (8.58 + 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.28iT - 97T^{2} \)
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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

−8.869848979154630689523325659282, −8.350626100704541708581253454059, −7.26948523850726532319105970748, −6.75053216827115372028483145138, −6.38065180453279700100030631500, −4.88521014523656680788646733606, −4.40798251989516312001559327706, −3.18901671710009576649675572452, −1.76609121591930164421487173516, −1.20279904139983686856081306433, 0.74238630826277402857386002496, 2.43646294642677357788210086912, 3.37469585768730165055579654243, 4.27214880295442975718057386428, 5.09528440653785926610085096928, 5.99144500400028279706647128710, 6.35621874680650341385208583661, 7.86631956390206811883877332421, 8.443405230429543048404843044076, 9.281514035558374820688489501787

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