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Properties

Label 2-2100-35.4-c1-0-21
Degree $2$
Conductor $2100$
Sign $-0.867 + 0.497i$
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  + (0.866 + 0.5i)3-s + (−2.29 + 1.32i)7-s + (0.499 + 0.866i)9-s + (−1.82 + 3.15i)11-s − 2.64i·13-s + (−3.15 − 1.82i)17-s + (−1.14 − 1.98i)19-s − 2.64·21-s + (3.15 − 1.82i)23-s + 0.999i·27-s − 2.35·29-s + (−3.14 + 5.44i)31-s + (−3.15 + 1.82i)33-s + (−4.02 + 2.32i)37-s + (1.32 − 2.29i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.866 + 0.499i)7-s + (0.166 + 0.288i)9-s + (−0.549 + 0.951i)11-s − 0.733i·13-s + (−0.765 − 0.442i)17-s + (−0.262 − 0.455i)19-s − 0.577·21-s + (0.658 − 0.380i)23-s + 0.192i·27-s − 0.437·29-s + (−0.564 + 0.978i)31-s + (−0.549 + 0.317i)33-s + (−0.661 + 0.381i)37-s + (0.211 − 0.366i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02832397113\)
\(L(\frac12)\) \(\approx\) \(0.02832397113\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (2.29 - 1.32i)T \)
good11 \( 1 + (1.82 - 3.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.64iT - 13T^{2} \)
17 \( 1 + (3.15 + 1.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.14 + 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.15 + 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + (3.14 - 5.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.02 - 2.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 9.93iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.31 + 3.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.46 - 4.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.02 - 2.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.35T + 71T^{2} \)
73 \( 1 + (11.4 + 6.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 + 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.93iT - 83T^{2} \)
89 \( 1 + (-6.11 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.943122185963691291652673095009, −8.127134137239610270545576551515, −7.10714328994551064887686295056, −6.66044423353524595627457508243, −5.36794555991010014477367621045, −4.86201197391625604207681360868, −3.65277124490269439141497864637, −2.85835039908857903047904890023, −2.00275000619171376666847141198, −0.008632333355186146999389222259, 1.55390705647446707241383391643, 2.76178731041876918394259924807, 3.57866634115695030525852885542, 4.37142775461334806090805486772, 5.63686956026456483029806208542, 6.39177147630330205668324913881, 7.08946893048632183386035451436, 7.889255391439776612496732011592, 8.694599615752417670564654672724, 9.320826619153902055439162858115

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