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Properties

Label 2-2904-264.227-c0-0-9
Degree $2$
Conductor $2904$
Sign $0.382 + 0.924i$
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  + (−0.587 + 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (1.14 − 0.831i)5-s + (0.587 + 0.809i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 1.41i·10-s − 12-s + (−0.437 − 1.34i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.309i)18-s + (−1.34 − 0.437i)19-s + (−1.14 − 0.831i)20-s + 1.41·23-s + (0.587 − 0.809i)24-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (1.14 − 0.831i)5-s + (0.587 + 0.809i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 1.41i·10-s − 12-s + (−0.437 − 1.34i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.309i)18-s + (−1.34 − 0.437i)19-s + (−1.14 − 0.831i)20-s + 1.41·23-s + (0.587 − 0.809i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $0.382 + 0.924i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (2339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ 0.382 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151861810\)
\(L(\frac12)\) \(\approx\) \(1.151861810\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + (-1.90 + 0.618i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−8.706209245623146694839601935784, −8.251524467286615165033418588970, −7.21866174980363129673159849855, −6.57445784304408836176779602871, −6.00996042017534590260840948610, −5.21142234481457184374167396116, −4.43848419620024526808166823315, −2.72085541778123865681002544806, −1.80946614204173083362702111318, −0.883184408390101345609889964756, 1.63873857338055313798199239080, 2.72018056300665611419873335559, 3.07581289270003134406013285697, 4.27002583425285094681555077206, 4.97160473374537746908435264659, 6.10854810176545710476288005899, 6.81570666519004977744612312745, 7.87333382896842519874676223069, 8.721713665192099247994912436383, 9.153142128451332845640656199968

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