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Properties

Label 2-2904-264.53-c0-0-0
Degree $2$
Conductor $2904$
Sign $-0.780 - 0.625i$
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  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)5-s + (−0.309 − 0.951i)6-s + (−1.30 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s − 0.618·10-s + 0.999·12-s + (1.30 − 0.951i)14-s + (−0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (0.190 − 0.587i)20-s + 1.61·21-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)5-s + (−0.309 − 0.951i)6-s + (−1.30 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s − 0.618·10-s + 0.999·12-s + (1.30 − 0.951i)14-s + (−0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (0.190 − 0.587i)20-s + 1.61·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (2429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5077227141\)
\(L(\frac12)\) \(\approx\) \(0.5077227141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.329265163839272465413502797026, −8.591176185268134469663241200289, −7.35251393403190727649803216361, −6.84374109855784419613448462949, −6.38909971093178674825504143591, −5.61667875483803772336097970534, −4.71236815440971210619974039215, −3.89457508207798164741372843394, −3.08552963213523406162191327326, −1.00321606474135885148749929693, 0.50330085827716811470284126643, 1.86059872132936677723868856270, 2.68218124539803972587152785726, 3.72926127812730658063568578177, 4.83608130307080750822286516719, 5.53880179441906736476548437637, 6.27290111324936187304626489887, 7.13171476914666459098543010497, 8.113980780538255758998671909485, 8.772545376674682325930239956731

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