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Properties

Label 2-2100-105.44-c0-0-0
Degree $2$
Conductor $2100$
Sign $-0.123 - 0.992i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
Motivic weight $0$
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 − 0.5i)7-s + (0.499 − 0.866i)9-s + i·13-s + (−0.5 + 0.866i)19-s + 0.999·21-s + 0.999i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + i·43-s + (0.499 + 0.866i)49-s − 0.999i·57-s + (−1 + 1.73i)61-s + (−0.866 + 0.499i)63-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + i·13-s + (−0.5 + 0.866i)19-s + 0.999·21-s + 0.999i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + i·43-s + (0.499 + 0.866i)49-s − 0.999i·57-s + (−1 + 1.73i)61-s + (−0.866 + 0.499i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.123 - 0.992i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ -0.123 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6041065809\)
\(L(\frac12)\) \(\approx\) \(0.6041065809\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−9.604013875876003089899530782979, −8.982465911952960256425440547803, −7.88738978252976762985000221614, −6.86165867065449479205639810100, −6.41065761858673800888353973442, −5.63806671498077390742563402673, −4.51351550238131101933914283167, −4.00434633353702026105377353524, −2.94492659033619765688171032502, −1.32623214897286410877514297542, 0.50978250457163993560915347559, 2.14422357075714411005594239815, 3.08942363672786550673663067584, 4.33604289263472834484335451662, 5.29065708075221113066669982200, 5.99184030846938665746950704856, 6.58905105779612545716608462741, 7.43977419015765692013445035156, 8.194520885035609358322228739972, 9.146082309823931869227042120002

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