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Properties

Label 2-2904-264.131-c0-0-10
Degree $2$
Conductor $2904$
Sign $0.846 - 0.531i$
Analytic cond. $1.44928$
Root an. cond. $1.20386$
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  + 2-s + (0.809 + 0.587i)3-s + 4-s + (0.809 + 0.587i)6-s + 8-s + (0.309 + 0.951i)9-s + (0.809 + 0.587i)12-s + 16-s − 0.618·17-s + (0.309 + 0.951i)18-s − 1.90i·19-s + (0.809 + 0.587i)24-s − 25-s + (−0.309 + 0.951i)27-s + 32-s + ⋯
L(s)  = 1  + 2-s + (0.809 + 0.587i)3-s + 4-s + (0.809 + 0.587i)6-s + 8-s + (0.309 + 0.951i)9-s + (0.809 + 0.587i)12-s + 16-s − 0.618·17-s + (0.309 + 0.951i)18-s − 1.90i·19-s + (0.809 + 0.587i)24-s − 25-s + (−0.309 + 0.951i)27-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $0.846 - 0.531i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ 0.846 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.060400650\)
\(L(\frac12)\) \(\approx\) \(3.060400650\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 0.618T + T^{2} \)
19 \( 1 + 1.90iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.90iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.17iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.61T + T^{2} \)
89 \( 1 + 1.17iT - T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−8.986778143140441684409394940057, −8.248896937012487588009266698084, −7.38836370129753746275059094868, −6.76884288586580370175398440257, −5.80588683070425051224774981511, −4.80203901390366524401020166166, −4.44434108699216036924102882372, −3.40125539772043001985547309886, −2.71928445071636869111669184865, −1.80842415462869066372663511175, 1.60893736557813558853162853943, 2.25556656906246645387209919305, 3.47370423002229540360480725955, 3.83971870579652064937489476693, 4.97089376675836681996000090630, 5.92314304229490559764109359566, 6.54018525288040467206012874065, 7.30835624315002558532209632218, 8.039341252445067064861702120504, 8.570901883012590844458436069985

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