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Properties

Label 2-3300-55.32-c1-0-21
Degree $2$
Conductor $3300$
Sign $0.0587 + 0.998i$
Analytic cond. $26.3506$
Root an. cond. $5.13328$
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  + (0.707 − 0.707i)3-s + (−3.37 + 3.37i)7-s − 1.00i·9-s + (−3.25 + 0.626i)11-s + (−1.72 − 1.72i)13-s + (−0.580 + 0.580i)17-s + 4.50·19-s + 4.77i·21-s + (3.65 − 3.65i)23-s + (−0.707 − 0.707i)27-s − 0.725·29-s + 6.16·31-s + (−1.86 + 2.74i)33-s + (0.978 + 0.978i)37-s − 2.43·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−1.27 + 1.27i)7-s − 0.333i·9-s + (−0.982 + 0.188i)11-s + (−0.477 − 0.477i)13-s + (−0.140 + 0.140i)17-s + 1.03·19-s + 1.04i·21-s + (0.761 − 0.761i)23-s + (−0.136 − 0.136i)27-s − 0.134·29-s + 1.10·31-s + (−0.323 + 0.477i)33-s + (0.160 + 0.160i)37-s − 0.390·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0587 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0587 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $0.0587 + 0.998i$
Analytic conductor: \(26.3506\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3300} (1957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3300,\ (\ :1/2),\ 0.0587 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086092528\)
\(L(\frac12)\) \(\approx\) \(1.086092528\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
11 \( 1 + (3.25 - 0.626i)T \)
good7 \( 1 + (3.37 - 3.37i)T - 7iT^{2} \)
13 \( 1 + (1.72 + 1.72i)T + 13iT^{2} \)
17 \( 1 + (0.580 - 0.580i)T - 17iT^{2} \)
19 \( 1 - 4.50T + 19T^{2} \)
23 \( 1 + (-3.65 + 3.65i)T - 23iT^{2} \)
29 \( 1 + 0.725T + 29T^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 + (-0.978 - 0.978i)T + 37iT^{2} \)
41 \( 1 - 8.99iT - 41T^{2} \)
43 \( 1 + (0.936 + 0.936i)T + 43iT^{2} \)
47 \( 1 + (8.25 + 8.25i)T + 47iT^{2} \)
53 \( 1 + (0.742 - 0.742i)T - 53iT^{2} \)
59 \( 1 + 9.89iT - 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + (1.66 + 1.66i)T + 67iT^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + (8.52 + 8.52i)T + 73iT^{2} \)
79 \( 1 - 8.00T + 79T^{2} \)
83 \( 1 + (-8.89 - 8.89i)T + 83iT^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (1.29 + 1.29i)T + 97iT^{2} \)
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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

−8.318844261380905874606842577714, −7.911545304113664477618525700073, −6.82711272220384409100240959947, −6.35675766471750131895155908679, −5.41784882129940412720116170265, −4.81059922262964862807195314931, −3.21619933688788133001748191179, −2.95166441048393650381894062980, −2.03235219595051169021070934540, −0.35728875288249951361510408901, 1.00108414652067369839228061481, 2.62660064973976337570437438139, 3.24555110886984295333727901713, 4.05434465253282300564271694294, 4.88385630526675690433316221330, 5.75874467214732892248050039999, 6.77577720427114704741958170401, 7.34719157755408483678541141033, 7.912447428785384444933714725593, 9.032032210473385040692332690192

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