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Properties

Label 2-2100-35.4-c1-0-20
Degree $2$
Conductor $2100$
Sign $-0.965 + 0.261i$
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 + (−2.09 − 1.62i)7-s + (0.499 + 0.866i)9-s + (2.12 − 3.67i)11-s + 3.24i·13-s + (3.67 + 2.12i)17-s + (−3.5 − 6.06i)19-s + (0.999 + 2.44i)21-s + (−3.67 + 2.12i)23-s − 0.999i·27-s + 1.75·29-s + (4.74 − 8.21i)31-s + (−3.67 + 2.12i)33-s + (−2.80 + 1.62i)37-s + (1.62 − 2.80i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.790 − 0.612i)7-s + (0.166 + 0.288i)9-s + (0.639 − 1.10i)11-s + 0.899i·13-s + (0.891 + 0.514i)17-s + (−0.802 − 1.39i)19-s + (0.218 + 0.534i)21-s + (−0.766 + 0.442i)23-s − 0.192i·27-s + 0.326·29-s + (0.851 − 1.47i)31-s + (−0.639 + 0.369i)33-s + (−0.461 + 0.266i)37-s + (0.259 − 0.449i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.965 + 0.261i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5781395782\)
\(L(\frac12)\) \(\approx\) \(0.5781395782\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 + 1.62i)T \)
good11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.24iT - 13T^{2} \)
17 \( 1 + (-3.67 - 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 + (-4.74 + 8.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.80 - 1.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 3.24iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.34 + 4.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.54 + 2.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (8.00 + 4.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.485iT - 97T^{2} \)
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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

−8.793364286604943046367884413263, −7.916952845236816596925516783217, −7.02411731891449760584952626833, −6.31984258025948213021647615152, −5.89896022844645782682218204967, −4.58396638606134973895844979694, −3.86400721491321422875333991204, −2.86043950724343263632848795746, −1.44445726164703261822669429636, −0.22682634731559279361046729131, 1.48598821670041611753944954311, 2.80980008860033721471333605847, 3.73099737404411196638279110975, 4.67187287439765757960296905309, 5.58671142815880160266788709739, 6.23356255611434827130998790557, 6.97690959806518395312362222586, 7.937396784990947758729809089516, 8.752925392650095840157204029219, 9.640514644207373397072143679400

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