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Properties

Label 2-2100-21.5-c1-0-37
Degree $2$
Conductor $2100$
Sign $-0.0543 + 0.998i$
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  + (−1.28 + 1.16i)3-s + (−1.63 + 2.08i)7-s + (0.287 − 2.98i)9-s + (5.09 − 2.93i)11-s + 3.54i·13-s + (−1.55 − 2.69i)17-s + (−3.58 − 2.06i)19-s + (−0.331 − 4.57i)21-s + (−6.19 − 3.57i)23-s + (3.10 + 4.16i)27-s + 6.84i·29-s + (−3.90 + 2.25i)31-s + (−3.10 + 9.69i)33-s + (0.986 − 1.70i)37-s + (−4.12 − 4.53i)39-s + ⋯
L(s)  = 1  + (−0.740 + 0.672i)3-s + (−0.617 + 0.786i)7-s + (0.0959 − 0.995i)9-s + (1.53 − 0.886i)11-s + 0.981i·13-s + (−0.376 − 0.652i)17-s + (−0.822 − 0.474i)19-s + (−0.0723 − 0.997i)21-s + (−1.29 − 0.745i)23-s + (0.598 + 0.801i)27-s + 1.27i·29-s + (−0.701 + 0.404i)31-s + (−0.540 + 1.68i)33-s + (0.162 − 0.281i)37-s + (−0.660 − 0.726i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0543 + 0.998i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0543 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4561482952\)
\(L(\frac12)\) \(\approx\) \(0.4561482952\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.16i)T \)
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

−8.957844945235284980495835106227, −8.594445590598170708547394369223, −6.84425182663909739621802423217, −6.55896779486095530529387213627, −5.82561111144655823857460515539, −4.86102456293399914658725573916, −4.00184363783635017904077325489, −3.25836180155172301086757731657, −1.84286324354131982643794010600, −0.18894077549897064041587062304, 1.21835689552313812597313746753, 2.19970696390631125734119047539, 3.81733950544275806669371130919, 4.25801273111391182055487073651, 5.58515850679683672898327421608, 6.28038839089619093033333817436, 6.83863016366620081990586886386, 7.63245447572404694160801017271, 8.321477718802874283019521168708, 9.472095363071843199215304113222

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