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Properties

Label 2-2100-105.89-c1-0-19
Degree $2$
Conductor $2100$
Sign $0.548 - 0.836i$
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 + (1.93 − 1.80i)7-s + (0.284 + 2.98i)9-s + (−4.05 + 2.34i)11-s + 2.18·13-s + (6.49 − 3.74i)17-s + (0.638 + 0.368i)19-s + (4.58 − 0.0477i)21-s + (−4.03 + 6.99i)23-s + (−3.11 + 4.15i)27-s + 1.15i·29-s + (8.95 − 5.16i)31-s + (−7.92 − 1.72i)33-s + (3.99 + 2.30i)37-s + (2.80 + 2.55i)39-s + ⋯
L(s)  = 1  + (0.739 + 0.672i)3-s + (0.732 − 0.680i)7-s + (0.0947 + 0.995i)9-s + (−1.22 + 0.706i)11-s + 0.607·13-s + (1.57 − 0.909i)17-s + (0.146 + 0.0845i)19-s + (0.999 − 0.0104i)21-s + (−0.842 + 1.45i)23-s + (−0.599 + 0.800i)27-s + 0.214i·29-s + (1.60 − 0.928i)31-s + (−1.37 − 0.300i)33-s + (0.657 + 0.379i)37-s + (0.449 + 0.408i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.548 - 0.836i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.548 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.502870443\)
\(L(\frac12)\) \(\approx\) \(2.502870443\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.80i)T \)
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

−9.401646783375780344050516729623, −8.195532362116701027545550647138, −7.82125839777493354454899870338, −7.32161329607693812717146596928, −5.83826777342852720969816868331, −5.05225199818843980740406621155, −4.37071741131716588734970490981, −3.42130951985924492388696314184, −2.54094736996629361009720615600, −1.29131083757597423897413661223, 0.915972218227020901029927279129, 2.12456126299081426402701792740, 2.92691656501571100045724678041, 3.85795980856838225564785189502, 5.10647688293082404358611877135, 5.88789395902228561517824756255, 6.58881310979125701883303961804, 7.82043325647567715146498757673, 8.287008024791159160637284821605, 8.488138819469276921458231121552

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