Properties

Label 2-2100-105.89-c1-0-46
Degree $2$
Conductor $2100$
Sign $-0.931 - 0.363i$
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  + (0.348 − 1.69i)3-s + (2.41 − 1.08i)7-s + (−2.75 − 1.18i)9-s + (1.17 − 0.675i)11-s − 4.94·13-s + (−4.97 + 2.87i)17-s + (−2.84 − 1.64i)19-s + (−0.993 − 4.47i)21-s + (−2.50 + 4.33i)23-s + (−2.96 + 4.26i)27-s − 5.68i·29-s + (−2.45 + 1.41i)31-s + (−0.738 − 2.22i)33-s + (−3.33 − 1.92i)37-s + (−1.72 + 8.38i)39-s + ⋯
L(s)  = 1  + (0.201 − 0.979i)3-s + (0.912 − 0.409i)7-s + (−0.918 − 0.394i)9-s + (0.353 − 0.203i)11-s − 1.37·13-s + (−1.20 + 0.696i)17-s + (−0.653 − 0.377i)19-s + (−0.216 − 0.976i)21-s + (−0.521 + 0.903i)23-s + (−0.571 + 0.820i)27-s − 1.05i·29-s + (−0.440 + 0.254i)31-s + (−0.128 − 0.386i)33-s + (−0.548 − 0.316i)37-s + (−0.276 + 1.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.931 - 0.363i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.931 - 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6094275201\)
\(L(\frac12)\) \(\approx\) \(0.6094275201\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.348 + 1.69i)T \)
5 \( 1 \)
7 \( 1 + (-2.41 + 1.08i)T \)
good11 \( 1 + (-1.17 + 0.675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 + (4.97 - 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 + 1.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.50 - 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.68iT - 29T^{2} \)
31 \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.33 + 1.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 + 4.06iT - 43T^{2} \)
47 \( 1 + (4.92 + 2.84i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.730 + 1.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.34 - 7.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 0.954i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.36 + 2.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.38iT - 71T^{2} \)
73 \( 1 + (8.16 + 14.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.41 + 4.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.6iT - 83T^{2} \)
89 \( 1 + (-8.08 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−8.537502314991344939563775801079, −7.84654674034504351265468520161, −7.16536144055851124123776192535, −6.51598175582438873444984594129, −5.54099416868740265887547637053, −4.62306662225453361270555636227, −3.68934400551722773665103611834, −2.32214909364244812240514137728, −1.72805186376005678009918075727, −0.18722821808517589347695079053, 1.98519323367401250468119445457, 2.73219549132605667757552574187, 4.01643059366347546241869728980, 4.77216288176202101604777583671, 5.19892909070416345361564478833, 6.37370591964470076818627890253, 7.26762690860384662347247154354, 8.252297898945881119201121265187, 8.763187148617210379919421565613, 9.529189457486304318821365520998

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