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Properties

Label 2-2100-21.17-c1-0-38
Degree $2$
Conductor $2100$
Sign $0.997 + 0.0633i$
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  + (1.5 + 0.866i)3-s + (2.5 − 0.866i)7-s + (1.5 + 2.59i)9-s − 5.19i·13-s + (7.5 − 4.33i)19-s + (4.5 + 0.866i)21-s + 5.19i·27-s + (−9 − 5.19i)31-s + (0.5 + 0.866i)37-s + (4.5 − 7.79i)39-s + 8·43-s + (5.5 − 4.33i)49-s + 15·57-s + (−13.5 + 7.79i)61-s + (6 + 5.19i)63-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.944 − 0.327i)7-s + (0.5 + 0.866i)9-s − 1.44i·13-s + (1.72 − 0.993i)19-s + (0.981 + 0.188i)21-s + 0.999i·27-s + (−1.61 − 0.933i)31-s + (0.0821 + 0.142i)37-s + (0.720 − 1.24i)39-s + 1.21·43-s + (0.785 − 0.618i)49-s + 1.98·57-s + (−1.72 + 0.997i)61-s + (0.755 + 0.654i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.843187574\)
\(L(\frac12)\) \(\approx\) \(2.843187574\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
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 \( 1 + (9 + 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
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 \( 1 + (13.5 - 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-13.5 - 7.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−9.215293485749154785716883046137, −8.197559621307013370347202214462, −7.69745725982321426073260772789, −7.14390614218787308485474263900, −5.58475424862193069708378679908, −5.10429327590881580721646497541, −4.12165593101399199337932657434, −3.24766862070557799029412032541, −2.37247346927163917509529778181, −1.04006884104734251891020201187, 1.37978528764240908399716322552, 2.02668303173117687912630079575, 3.23656042104179784030453871975, 4.08591098968372334681320559963, 5.08126954872234791849929113241, 6.00551208795012122467782802700, 7.03763539329666039795409260529, 7.60193180788418036503028117559, 8.269162957708945794247702833254, 9.290814512440635611877066157263

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