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Properties

Label 2-2100-21.5-c1-0-26
Degree $2$
Conductor $2100$
Sign $0.553 + 0.832i$
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  − 1.73i·3-s + (−2 + 1.73i)7-s − 2.99·9-s + (4.5 − 2.59i)11-s + (1.5 + 2.59i)17-s + (1.5 + 0.866i)19-s + (2.99 + 3.46i)21-s + (4.5 + 2.59i)23-s + 5.19i·27-s + (−1.5 + 0.866i)31-s + (−4.5 − 7.79i)33-s + (3.5 − 6.06i)37-s + 6·41-s − 4·43-s + (1.5 − 2.59i)47-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.755 + 0.654i)7-s − 0.999·9-s + (1.35 − 0.783i)11-s + (0.363 + 0.630i)17-s + (0.344 + 0.198i)19-s + (0.654 + 0.755i)21-s + (0.938 + 0.541i)23-s + 0.999i·27-s + (−0.269 + 0.155i)31-s + (−0.783 − 1.35i)33-s + (0.575 − 0.996i)37-s + 0.937·41-s − 0.609·43-s + (0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.674701335\)
\(L(\frac12)\) \(\approx\) \(1.674701335\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 2.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.109973881949452515216461718691, −8.199512873309448616974855043970, −7.37265906716638787077052161511, −6.51246513372365782856174250116, −6.04504927791908449143050728634, −5.28590064180226793458155389333, −3.74965120638363255222258218215, −3.11189083458223658299959434566, −1.92369388143267918635472910761, −0.804882858369840061750866089629, 0.955467878460314019854935514363, 2.67169347977523904264749237920, 3.56265118176837839126624826247, 4.30540228059946861434333766714, 5.01333799700500799458235928081, 6.13336287428775606616437436484, 6.82304698227064969233697271419, 7.59957154936473298936435109538, 8.744383739496235867106539334545, 9.452295183712603916768117369002

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