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Properties

Label 2-2100-105.89-c1-0-24
Degree $2$
Conductor $2100$
Sign $0.263 + 0.964i$
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 − 1.5i)3-s + (−0.866 − 2.5i)7-s + (−1.5 + 2.59i)9-s + 5.19·13-s + (7.5 + 4.33i)19-s + (−3 + 3.46i)21-s + 5.19·27-s + (−1.5 + 0.866i)31-s + (9.52 + 5.5i)37-s + (−4.5 − 7.79i)39-s − 13i·43-s + (−5.5 + 4.33i)49-s − 15i·57-s + (−6 − 3.46i)61-s + (7.79 + 1.5i)63-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.327 − 0.944i)7-s + (−0.5 + 0.866i)9-s + 1.44·13-s + (1.72 + 0.993i)19-s + (−0.654 + 0.755i)21-s + 1.00·27-s + (−0.269 + 0.155i)31-s + (1.56 + 0.904i)37-s + (−0.720 − 1.24i)39-s − 1.98i·43-s + (−0.785 + 0.618i)49-s − 1.98i·57-s + (−0.768 − 0.443i)61-s + (0.981 + 0.188i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.263 + 0.964i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.512913501\)
\(L(\frac12)\) \(\approx\) \(1.512913501\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19T + 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 + 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-0.866 - 1.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−8.826360779850188091573857701204, −7.903438258179513292181081631426, −7.45212554261275645271708449170, −6.54370551012614037540935593046, −5.95685369466019610939453945330, −5.10517585265400638323226588872, −3.91036923916458468183984983633, −3.12612737638943727670198949984, −1.61824169594713803205439979276, −0.78073866798496844434912338139, 0.997681612093199119435921138460, 2.72018937318375139922238768973, 3.46596596449212001358297441026, 4.45303086233147143445711943289, 5.39207686229490037829971772003, 5.94042228741082215279408569608, 6.66900068397730155300613618124, 7.83368606942457131764828082398, 8.726458747773301857994102007795, 9.412198378309970418898322155819

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