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Properties

Label 2-2100-21.17-c1-0-49
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 − 1.73i)7-s + (1.5 − 2.59i)9-s + (−4.5 − 2.59i)11-s + (−1.5 + 2.59i)17-s + (1.5 − 0.866i)19-s + (−4.5 − 0.866i)21-s + (−4.5 + 2.59i)23-s − 5.19i·27-s + (−1.5 − 0.866i)31-s − 9·33-s + (3.5 + 6.06i)37-s − 6·41-s − 4·43-s + (−1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.755 − 0.654i)7-s + (0.5 − 0.866i)9-s + (−1.35 − 0.783i)11-s + (−0.363 + 0.630i)17-s + (0.344 − 0.198i)19-s + (−0.981 − 0.188i)21-s + (−0.938 + 0.541i)23-s − 0.999i·27-s + (−0.269 − 0.155i)31-s − 1.56·33-s + (0.575 + 0.996i)37-s − 0.937·41-s − 0.609·43-s + (−0.218 − 0.378i)47-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(0.7593682700\)
\(L(\frac12)\) \(\approx\) \(0.7593682700\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
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

−8.490452255505808785718035669142, −8.010181908635273792347359693350, −7.27463372978479410304515421520, −6.45990051916360542140979160010, −5.70210627268010743458853329204, −4.47412481066229869345731858218, −3.44921893162422662968174581876, −2.90270162116892277370524798952, −1.68585540233826341418040820739, −0.21524385562713649153125238510, 2.07864189985384542470700456706, 2.71378840005693220288663643447, 3.61382408954295498733610555651, 4.69494805749191781783652189239, 5.33753203636196163410285946596, 6.39648405519992540403969591284, 7.39116521846833525450400527146, 7.984196671378131689082731245438, 8.801955347775190986147944562992, 9.544966386232830624155382989274

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