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Properties

Label 2-2100-21.17-c1-0-22
Degree $2$
Conductor $2100$
Sign $0.643 + 0.765i$
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  + (−0.884 − 1.48i)3-s + (2.64 − 0.0973i)7-s + (−1.43 + 2.63i)9-s + (−2.62 − 1.51i)11-s + 2.31i·13-s + (−2.59 + 4.49i)17-s + (5.58 − 3.22i)19-s + (−2.48 − 3.85i)21-s + (4.21 − 2.43i)23-s + (5.19 − 0.194i)27-s − 3.48i·29-s + (1.16 + 0.673i)31-s + (0.0657 + 5.25i)33-s + (1.40 + 2.43i)37-s + (3.45 − 2.05i)39-s + ⋯
L(s)  = 1  + (−0.510 − 0.859i)3-s + (0.999 − 0.0368i)7-s + (−0.478 + 0.878i)9-s + (−0.792 − 0.457i)11-s + 0.642i·13-s + (−0.629 + 1.09i)17-s + (1.28 − 0.739i)19-s + (−0.542 − 0.840i)21-s + (0.879 − 0.507i)23-s + (0.999 − 0.0374i)27-s − 0.647i·29-s + (0.209 + 0.120i)31-s + (0.0114 + 0.915i)33-s + (0.231 + 0.400i)37-s + (0.552 − 0.328i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.643 + 0.765i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.547106991\)
\(L(\frac12)\) \(\approx\) \(1.547106991\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.884 + 1.48i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.0973i)T \)
good11 \( 1 + (2.62 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.31iT - 13T^{2} \)
17 \( 1 + (2.59 - 4.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.58 + 3.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.21 + 2.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.48iT - 29T^{2} \)
31 \( 1 + (-1.16 - 0.673i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.40 - 2.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.06T + 41T^{2} \)
43 \( 1 + 2.42T + 43T^{2} \)
47 \( 1 + (-1.90 - 3.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.11 - 3.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.18 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 + 6.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.14 - 3.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.08iT - 71T^{2} \)
73 \( 1 + (0.132 + 0.0763i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.04 - 5.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (7.10 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.63iT - 97T^{2} \)
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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

−8.619597127483819541641967784100, −8.278026296784976608458044623957, −7.36238445415063379647943132973, −6.76768457413975392694302134606, −5.78988588456738034771551033776, −5.11535787191735450714963380063, −4.31809682957240809172730364427, −2.85194061902294984892618706409, −1.92187040553060960296685306574, −0.804713943664733048766698913347, 0.924963857193860783770481622429, 2.49028702057948147422747756648, 3.48465793545007895368948285427, 4.56107533971498364528082656040, 5.26267720334701286300370538021, 5.60555960726240182953073659300, 7.01276482529135995078490258115, 7.60118270084630247511939516824, 8.552096341461800871126884539441, 9.271405287334550370267353252370

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