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Properties

Label 2-2904-264.125-c0-0-3
Degree $2$
Conductor $2904$
Sign $0.0694 - 0.997i$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)5-s + (0.809 − 0.587i)6-s + (0.5 + 1.53i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 0.618·10-s + 12-s + (−0.5 + 1.53i)14-s + (0.190 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.363i)20-s + 1.61·21-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)5-s + (0.809 − 0.587i)6-s + (0.5 + 1.53i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 0.618·10-s + 12-s + (−0.5 + 1.53i)14-s + (0.190 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.363i)20-s + 1.61·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (2501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ 0.0694 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923985954\)
\(L(\frac12)\) \(\approx\) \(1.923985954\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.660578338685978430451915727884, −8.266202866568331053158168914211, −7.55668500107926261302249864543, −6.83931237584496531523942553739, −6.06208593548660118367147607139, −5.49536131013422232554148009214, −4.57834304489644056139325592608, −3.38601364471959348270430861091, −2.71240870706274125933986486446, −1.86230757303474009071058746396, 0.906541681093607830601287172479, 2.32999837619646567804062271915, 3.44215584414383891548248032036, 4.12479585282174260619067104896, 4.52423578025727434564383808139, 5.29252431984993082153284443868, 6.32169700409563256566499361523, 7.31358237875837364213372461501, 8.033250046910044314400086032059, 8.864888076186605110637706779830

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