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Properties

Label 2-2100-35.4-c1-0-5
Degree $2$
Conductor $2100$
Sign $-0.556 - 0.830i$
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 + (−0.866 + 2.5i)7-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s + 7i·13-s + (−5.19 − 3i)17-s + (1.5 + 2.59i)19-s + (−2 + 1.73i)21-s + (−1.73 + i)23-s + 0.999i·27-s + 2·29-s + (−3.5 + 6.06i)31-s + (3.46 − 1.99i)33-s + (6.06 − 3.5i)37-s + (−3.5 + 6.06i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.327 + 0.944i)7-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + 1.94i·13-s + (−1.26 − 0.727i)17-s + (0.344 + 0.596i)19-s + (−0.436 + 0.377i)21-s + (−0.361 + 0.208i)23-s + 0.192i·27-s + 0.371·29-s + (−0.628 + 1.08i)31-s + (0.603 − 0.348i)33-s + (0.996 − 0.575i)37-s + (−0.560 + 0.970i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.556 - 0.830i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.556 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.507179209\)
\(L(\frac12)\) \(\approx\) \(1.507179209\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 2.5i)T \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 7iT - 13T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (8.66 - 5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.92 - 4i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (12.9 + 7.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.134094986220568502730932383732, −8.892722997853041302448893233502, −8.041702445510955034239253050332, −6.81052193510428048660673918927, −6.41377950513616464026980102715, −5.36154220020574193797762323702, −4.41102754424863926746585231563, −3.58642653378470077180351096855, −2.61045979519073208635472264401, −1.63419916026507724319743651807, 0.47882458029656790425452158964, 1.82603767479830336944618780585, 2.93462618274489109843747650098, 3.86891073758060899551150726913, 4.60781787351233977898374433539, 5.74543774172179628518830117111, 6.74926029909613058076076064823, 7.21035495191030004535051887580, 8.088682898300201277909137853987, 8.680317738669968197998195263485

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