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Properties

Label 2-2100-7.2-c1-0-3
Degree $2$
Conductor $2100$
Sign $0.222 - 0.974i$
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  + (−0.5 − 0.866i)3-s + (−0.618 − 2.57i)7-s + (−0.499 + 0.866i)9-s + (−1.46 − 2.53i)11-s − 5.76·13-s + (−0.118 − 0.205i)17-s + (−2.37 + 4.10i)19-s + (−1.91 + 1.82i)21-s + (−3.38 + 5.86i)23-s + 0.999·27-s + 6.03·29-s + (5.26 + 9.12i)31-s + (−1.46 + 2.53i)33-s + (1.53 − 2.66i)37-s + (2.88 + 4.99i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.233 − 0.972i)7-s + (−0.166 + 0.288i)9-s + (−0.442 − 0.765i)11-s − 1.60·13-s + (−0.0287 − 0.0497i)17-s + (−0.543 + 0.941i)19-s + (−0.418 + 0.397i)21-s + (−0.705 + 1.22i)23-s + 0.192·27-s + 1.12·29-s + (0.946 + 1.63i)31-s + (−0.255 + 0.442i)33-s + (0.252 − 0.437i)37-s + (0.461 + 0.800i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.222 - 0.974i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.222 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5302444019\)
\(L(\frac12)\) \(\approx\) \(0.5302444019\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.618 + 2.57i)T \)
good11 \( 1 + (1.46 + 2.53i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 + (0.118 + 0.205i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.37 - 4.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.38 - 5.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 + (-5.26 - 9.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.53 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.06T + 41T^{2} \)
43 \( 1 - 9.79T + 43T^{2} \)
47 \( 1 + (1.85 - 3.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.50 + 9.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.65 - 8.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.06 + 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.12 - 1.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (1.89 + 3.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.00873 - 0.0151i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.30T + 83T^{2} \)
89 \( 1 + (8.97 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.31T + 97T^{2} \)
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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

−9.365191950251626458050742204652, −8.211108388097725665407925636221, −7.72530782126327242532492220154, −6.96585092075147717031018873989, −6.22113469337802591595654291671, −5.31942437984353903781554942624, −4.48004653612939698395768804373, −3.42577691210762991285219507817, −2.41851052857067901256964484489, −1.09313567449201455597215817009, 0.21116000745734928028831871049, 2.37861874310986930462719305457, 2.69748362653027053822257749712, 4.45086325608694112659865213405, 4.67547598243436850003641736192, 5.75742200456572497153967826897, 6.47243520319763012160800463830, 7.37257297323501149552273238766, 8.251638265094740252767556128102, 9.035439507695193142044509552309

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