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Properties

Label 2-2100-21.5-c1-0-8
Degree $2$
Conductor $2100$
Sign $0.389 - 0.921i$
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  + (−1.52 − 0.813i)3-s + (−0.567 − 2.58i)7-s + (1.67 + 2.48i)9-s + (0.793 − 0.457i)11-s + 4.31i·13-s + (0.268 + 0.465i)17-s + (1.12 + 0.651i)19-s + (−1.23 + 4.41i)21-s + (−6.91 − 3.99i)23-s + (−0.537 − 5.16i)27-s + 3.46i·29-s + (−5.56 + 3.21i)31-s + (−1.58 + 0.0547i)33-s + (−2.52 + 4.36i)37-s + (3.50 − 6.59i)39-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)3-s + (−0.214 − 0.976i)7-s + (0.558 + 0.829i)9-s + (0.239 − 0.138i)11-s + 1.19i·13-s + (0.0651 + 0.112i)17-s + (0.258 + 0.149i)19-s + (−0.269 + 0.962i)21-s + (−1.44 − 0.832i)23-s + (−0.103 − 0.994i)27-s + 0.642i·29-s + (−0.998 + 0.576i)31-s + (−0.275 + 0.00952i)33-s + (−0.414 + 0.718i)37-s + (0.561 − 1.05i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.389 - 0.921i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.389 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7277904786\)
\(L(\frac12)\) \(\approx\) \(0.7277904786\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 + 0.813i)T \)
5 \( 1 \)
7 \( 1 + (0.567 + 2.58i)T \)
good11 \( 1 + (-0.793 + 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.31iT - 13T^{2} \)
17 \( 1 + (-0.268 - 0.465i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 - 0.651i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.91 + 3.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.52 - 4.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 8.22T + 43T^{2} \)
47 \( 1 + (-2.17 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.70 + 0.984i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.15 - 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.38 + 4.84i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.05 - 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.943iT - 71T^{2} \)
73 \( 1 + (-8.85 + 5.11i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.71 - 9.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.35T + 83T^{2} \)
89 \( 1 + (0.874 - 1.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−9.335306417744018938383839008455, −8.352965994450424284487550989087, −7.49207451212540049097121836213, −6.80683629032313972721109306277, −6.30713160253537871748416805109, −5.31231985503470524860132096396, −4.38064987406405553261926616484, −3.69838719910154742917929688942, −2.12139482538565950342027596033, −1.09503857684963891529516147627, 0.32803478493072576920028752414, 1.94643221648613209339952365508, 3.22281255885104175938762542918, 4.07196682003126003254868696277, 5.19037383049115913762106603434, 5.72834062005273373006167749467, 6.30164912974539093010066563296, 7.41846727666647364244454715550, 8.150191777806580094240015239290, 9.252849984708691116284805260216

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