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Properties

Label 2-2100-21.20-c1-0-5
Degree $2$
Conductor $2100$
Sign $-0.987 - 0.156i$
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  + (1.30 + 1.14i)3-s + (−2.23 + 1.41i)7-s + (0.381 + 2.97i)9-s + 1.13i·11-s − 0.333i·13-s − 4.20·17-s − 1.95i·19-s + (−4.52 − 0.719i)21-s − 2.54i·23-s + (−2.90 + 4.30i)27-s + 8.49i·29-s + 5.11i·31-s + (−1.30 + 1.47i)33-s − 2.23·37-s + (0.381 − 0.434i)39-s + ⋯
L(s)  = 1  + (0.750 + 0.660i)3-s + (−0.845 + 0.534i)7-s + (0.127 + 0.991i)9-s + 0.342i·11-s − 0.0925i·13-s − 1.02·17-s − 0.448i·19-s + (−0.987 − 0.156i)21-s − 0.529i·23-s + (−0.559 + 0.828i)27-s + 1.57i·29-s + 0.918i·31-s + (−0.226 + 0.257i)33-s − 0.367·37-s + (0.0611 − 0.0695i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041187589\)
\(L(\frac12)\) \(\approx\) \(1.041187589\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 1.14i)T \)
5 \( 1 \)
7 \( 1 + (2.23 - 1.41i)T \)
good11 \( 1 - 1.13iT - 11T^{2} \)
13 \( 1 + 0.333iT - 13T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + 1.95iT - 19T^{2} \)
23 \( 1 + 2.54iT - 23T^{2} \)
29 \( 1 - 8.49iT - 29T^{2} \)
31 \( 1 - 5.11iT - 31T^{2} \)
37 \( 1 + 2.23T + 37T^{2} \)
41 \( 1 + 5.81T + 41T^{2} \)
43 \( 1 + 0.527T + 43T^{2} \)
47 \( 1 + 7.42T + 47T^{2} \)
53 \( 1 + 8.22iT - 53T^{2} \)
59 \( 1 + 3.59T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + 1.41iT - 73T^{2} \)
79 \( 1 - 1.47T + 79T^{2} \)
83 \( 1 + 5.81T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 - 9.69iT - 97T^{2} \)
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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

−9.193682684886679667894446807020, −8.952155069070821465447205740106, −8.159321426625591517116109972108, −7.05541488077840408706523009436, −6.50602919571286847976812543137, −5.26452163773520757920465562993, −4.65980952744232732851815440651, −3.54881228422352512148375556047, −2.88037497223375206323807620965, −1.88211536272839701291477373248, 0.30564773336567441793370285059, 1.72671266827040590539635879257, 2.78778117380935451637461718482, 3.64562653427307865386590278307, 4.41300305663007791281841264067, 5.87654043883431236272557924638, 6.46740707172971351204095643602, 7.21373030086141961672328032083, 7.952177819303967397434716139135, 8.662457043957059305333332985743

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