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Properties

Label 2-2100-21.5-c1-0-20
Degree $2$
Conductor $2100$
Sign $0.515 + 0.856i$
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  + (−0.990 − 1.42i)3-s + (−2.64 + 0.156i)7-s + (−1.03 + 2.81i)9-s + (−4.92 + 2.84i)11-s + 1.43i·13-s + (−1.62 − 2.81i)17-s + (5.43 + 3.13i)19-s + (2.83 + 3.59i)21-s + (0.884 + 0.510i)23-s + (5.02 − 1.31i)27-s − 4.95i·29-s + (−2.28 + 1.31i)31-s + (8.91 + 4.17i)33-s + (4.55 − 7.88i)37-s + (2.04 − 1.42i)39-s + ⋯
L(s)  = 1  + (−0.572 − 0.820i)3-s + (−0.998 + 0.0592i)7-s + (−0.345 + 0.938i)9-s + (−1.48 + 0.857i)11-s + 0.398i·13-s + (−0.393 − 0.682i)17-s + (1.24 + 0.719i)19-s + (0.619 + 0.784i)21-s + (0.184 + 0.106i)23-s + (0.967 − 0.253i)27-s − 0.920i·29-s + (−0.410 + 0.236i)31-s + (1.55 + 0.727i)33-s + (0.748 − 1.29i)37-s + (0.326 − 0.227i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.515 + 0.856i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8384275035\)
\(L(\frac12)\) \(\approx\) \(0.8384275035\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.990 + 1.42i)T \)
5 \( 1 \)
7 \( 1 + (2.64 - 0.156i)T \)
good11 \( 1 + (4.92 - 2.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.43iT - 13T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.43 - 3.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.884 - 0.510i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.95iT - 29T^{2} \)
31 \( 1 + (2.28 - 1.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.55 + 7.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.203T + 41T^{2} \)
43 \( 1 + 3.91T + 43T^{2} \)
47 \( 1 + (-5.76 + 9.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.21 + 5.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.739 + 1.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.62 - 4.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 - 4.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.08iT - 71T^{2} \)
73 \( 1 + (7.82 - 4.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.80 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-7.53 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−8.983082567831304995391843598553, −7.964067026336114234671470320999, −7.24149158787134546235971730545, −6.86351591466273531389412965091, −5.64243345108076099120552406363, −5.34678456995118729932909653147, −4.11286986431055427868765830587, −2.82090533573626164596648891097, −2.08513192822428582603365189578, −0.49524764239061691203616464322, 0.71876997479768436634685933123, 2.85285967710108199467183538149, 3.27605823409583947483645763151, 4.44150780691099571948510603490, 5.37028192801874077117318083145, 5.86269017051613711722055617387, 6.75059525572999877966231470552, 7.67879409701553920489964912029, 8.636637017520034568886277983655, 9.314461929301718650171683529920

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