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Properties

Label 2-2100-105.89-c1-0-30
Degree $2$
Conductor $2100$
Sign $-0.267 + 0.963i$
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  + (−0.368 − 1.69i)3-s + (−1.93 + 1.80i)7-s + (−2.72 + 1.24i)9-s + (4.05 − 2.34i)11-s − 2.18·13-s + (6.49 − 3.74i)17-s + (0.638 + 0.368i)19-s + (3.76 + 2.61i)21-s + (−4.03 + 6.99i)23-s + (3.11 + 4.15i)27-s − 1.15i·29-s + (8.95 − 5.16i)31-s + (−5.45 − 6.00i)33-s + (−3.99 − 2.30i)37-s + (0.806 + 3.70i)39-s + ⋯
L(s)  = 1  + (−0.212 − 0.977i)3-s + (−0.732 + 0.680i)7-s + (−0.909 + 0.415i)9-s + (1.22 − 0.706i)11-s − 0.607·13-s + (1.57 − 0.909i)17-s + (0.146 + 0.0845i)19-s + (0.820 + 0.571i)21-s + (−0.842 + 1.45i)23-s + (0.599 + 0.800i)27-s − 0.214i·29-s + (1.60 − 0.928i)31-s + (−0.950 − 1.04i)33-s + (−0.657 − 0.379i)37-s + (0.129 + 0.593i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.267 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.294878333\)
\(L(\frac12)\) \(\approx\) \(1.294878333\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.368 + 1.69i)T \)
5 \( 1 \)
7 \( 1 + (1.93 - 1.80i)T \)
good11 \( 1 + (-4.05 + 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + (-6.49 + 3.74i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.638 - 0.368i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.03 - 6.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.15iT - 29T^{2} \)
31 \( 1 + (-8.95 + 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.99 + 2.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 + 9.24iT - 43T^{2} \)
47 \( 1 + (7.52 + 4.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.06 - 7.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.48 + 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.13 - 2.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.19 + 0.691i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (0.122 + 0.211i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.79 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + (0.658 - 1.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.84T + 97T^{2} \)
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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

−8.878508377515368655428270299302, −7.968811894991504824691272573475, −7.32352363445914348497302539240, −6.45952878654558628276940634744, −5.83759947935216218010863428306, −5.19642583694244576947174627345, −3.66362620962439344225334032679, −2.93851585209417301570227989033, −1.78257261741004335221746017784, −0.54256834914415576572548644189, 1.14735797354979603457893272193, 2.82348137756031250744503549989, 3.72581195376183605640740677558, 4.35500584940085104806796523345, 5.19750591420892587474738827207, 6.37841703550313979211014425009, 6.66933674800398138257870884430, 7.916377637822711027092108597347, 8.623072725583835457355776463407, 9.757345139744571131400944404978

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