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Properties

Label 2-2904-264.125-c0-0-4
Degree $2$
Conductor $2904$
Sign $-0.642 + 0.766i$
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 + (1.30 − 0.951i)5-s + (−0.809 + 0.587i)6-s + (0.190 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + 12-s + (0.190 − 0.587i)14-s + (−0.499 − 1.53i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (1.30 + 0.951i)20-s + 0.618·21-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (1.30 − 0.951i)5-s + (−0.809 + 0.587i)6-s + (0.190 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + 12-s + (0.190 − 0.587i)14-s + (−0.499 − 1.53i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (1.30 + 0.951i)20-s + 0.618·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $-0.642 + 0.766i$
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: \((2,\ 2904,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136632046\)
\(L(\frac12)\) \(\approx\) \(1.136632046\)
\(L(1)\) 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 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.851963370391059605700703027880, −8.095338223446039458896747801951, −7.45910620315128204125035341757, −6.36263326559874682380639708191, −5.86081393909935451025327429439, −4.86628206165180422489739527460, −3.57884335047936022989625788259, −2.30175905030070490692213069544, −2.02191124598267743441681762608, −0.910882486956105059015886570375, 1.62986697657994633061680917106, 2.57657819590596686658398721146, 3.55037966813119820378290849990, 4.82512725306280755408863812860, 5.50134642175679727362391177396, 6.23581899080865946649017510179, 7.01756052500887949628620315460, 7.68479454030651748068632542903, 8.694873858472237477761501468220, 9.273869272835804848854721760451

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