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Properties

Label 2-2100-21.5-c1-0-13
Degree $2$
Conductor $2100$
Sign $-0.205 - 0.978i$
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 + (−0.5 + 2.59i)7-s + (1.5 − 2.59i)9-s − 1.73i·13-s + (4.5 + 2.59i)19-s + (−1.5 − 4.33i)21-s + 5.19i·27-s + (9 − 5.19i)31-s + (−5.5 + 9.52i)37-s + (1.49 + 2.59i)39-s + 8·43-s + (−6.5 − 2.59i)49-s − 9·57-s + (−7.5 − 4.33i)61-s + (6 + 5.19i)63-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.188 + 0.981i)7-s + (0.5 − 0.866i)9-s − 0.480i·13-s + (1.03 + 0.596i)19-s + (−0.327 − 0.944i)21-s + 0.999i·27-s + (1.61 − 0.933i)31-s + (−0.904 + 1.56i)37-s + (0.240 + 0.416i)39-s + 1.21·43-s + (−0.928 − 0.371i)49-s − 1.19·57-s + (−0.960 − 0.554i)61-s + (0.755 + 0.654i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.205 - 0.978i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086491110\)
\(L(\frac12)\) \(\approx\) \(1.086491110\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−9.581950899791057790654591589964, −8.603076792890712723112962449592, −7.83507468217309468198889740798, −6.76018892118948254980797444285, −6.01646777509454074820062745265, −5.41934086171437370720867397585, −4.65470576671331480995977807962, −3.57999959231947949951621920204, −2.64013528603393592114690897856, −1.10309200269745362228949041035, 0.52215734284593936210069224337, 1.58183247522925483768296780285, 2.95162090774115580860607682871, 4.15162543046267556401927410788, 4.87224034530790931437938025971, 5.77998359134134617036402373952, 6.65743859286813425118761954100, 7.21071891099072531343893674178, 7.82882955143568715616064285461, 8.918578397705236237280308743391

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