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Properties

Label 2-2904-264.227-c0-0-6
Degree $2$
Conductor $2904$
Sign $0.999 - 0.0237i$
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 + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 12-s + (−0.809 + 0.587i)16-s + (1.61 − 1.17i)17-s + (0.309 + 0.951i)18-s + (0.809 + 0.587i)24-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s − 2·34-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 12-s + (−0.809 + 0.587i)16-s + (1.61 − 1.17i)17-s + (0.309 + 0.951i)18-s + (0.809 + 0.587i)24-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s − 2·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $0.999 - 0.0237i$
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: \((2,\ 2904,\ (\ :0),\ 0.999 - 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7412043039\)
\(L(\frac12)\) \(\approx\) \(0.7412043039\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.236757956011575368608962759480, −8.417051539570077306541238255922, −7.57872989834050892959440604662, −6.90959031559866988212706523611, −5.71776169488531018458828533810, −5.09364348182514834238925263480, −3.92952870216302752149624766244, −3.36396619629787049174030663900, −2.39130927662032560313419602500, −0.865678770490946496922066828240, 0.977784251119320904949550260392, 1.90747178319055671876540594886, 3.03056681889302992777353340622, 4.47020245757565524155650822346, 5.59033175084657971336122191547, 5.99760421956874042952705176794, 6.73814340860450498667872654059, 7.58087366886331924560262696742, 8.083124759138922758552147151564, 8.611306007738283564870168723210

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