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Properties

Label 2-3300-55.43-c1-0-25
Degree $2$
Conductor $3300$
Sign $-0.155 + 0.987i$
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 + (−0.414 − 0.414i)7-s + 1.00i·9-s + (−3.18 − 0.941i)11-s + (3.69 − 3.69i)13-s + (0.270 + 0.270i)17-s + 7.48·19-s + 0.586i·21-s + (1.83 + 1.83i)23-s + (0.707 − 0.707i)27-s − 5.90·29-s − 1.59·31-s + (1.58 + 2.91i)33-s + (6.85 − 6.85i)37-s − 5.21·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.156 − 0.156i)7-s + 0.333i·9-s + (−0.958 − 0.283i)11-s + (1.02 − 1.02i)13-s + (0.0657 + 0.0657i)17-s + 1.71·19-s + 0.127i·21-s + (0.382 + 0.382i)23-s + (0.136 − 0.136i)27-s − 1.09·29-s − 0.286·31-s + (0.275 + 0.507i)33-s + (1.12 − 1.12i)37-s − 0.835·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.155 + 0.987i$
Analytic conductor: \(26.3506\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3300} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3300,\ (\ :1/2),\ -0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.311023703\)
\(L(\frac12)\) \(\approx\) \(1.311023703\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.18 + 0.941i)T \)
good7 \( 1 + (0.414 + 0.414i)T + 7iT^{2} \)
13 \( 1 + (-3.69 + 3.69i)T - 13iT^{2} \)
17 \( 1 + (-0.270 - 0.270i)T + 17iT^{2} \)
19 \( 1 - 7.48T + 19T^{2} \)
23 \( 1 + (-1.83 - 1.83i)T + 23iT^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 + (-6.85 + 6.85i)T - 37iT^{2} \)
41 \( 1 - 9.62iT - 41T^{2} \)
43 \( 1 + (8.26 - 8.26i)T - 43iT^{2} \)
47 \( 1 + (-2.66 + 2.66i)T - 47iT^{2} \)
53 \( 1 + (-5.53 - 5.53i)T + 53iT^{2} \)
59 \( 1 + 1.32iT - 59T^{2} \)
61 \( 1 + 4.30iT - 61T^{2} \)
67 \( 1 + (-7.04 + 7.04i)T - 67iT^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + (-1.83 + 1.83i)T - 73iT^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + (-9.58 + 9.58i)T - 83iT^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + (-8.42 + 8.42i)T - 97iT^{2} \)
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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

−8.177996395105467004020034890967, −7.71786388316088886269813821250, −7.06041001724323644266640384747, −5.91261573790515539096280316419, −5.63289449117300952527786667120, −4.76178758613370115851500856541, −3.48349505271097835489640775653, −2.93279925147679233208567061875, −1.53786441695754269304344250153, −0.49650448539230289547493487174, 1.09610186382678043017118425504, 2.36945151633437943877130034110, 3.42450813028045203126274679563, 4.16575179868679175448907150744, 5.21161841029814296141321289306, 5.60693262387483956477934304960, 6.61722166788715237314027283216, 7.27536137937722865401103369441, 8.104952424956833388086960301850, 9.018491914563853120596847024232

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