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Properties

Label 2-48e2-72.67-c0-0-0
Degree $2$
Conductor $2304$
Sign $0.422 + 0.906i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  − 3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)15-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)23-s − 27-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999·35-s + (0.866 + 0.5i)39-s + ⋯
L(s)  = 1  − 3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)15-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)23-s − 27-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999·35-s + (0.866 + 0.5i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.422 + 0.906i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.422 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4980383328\)
\(L(\frac12)\) \(\approx\) \(0.4980383328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
good5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.276897196556072667978914053556, −8.204575950650352902155448030968, −7.30750684561161779319103265124, −6.83203822227576207444984596141, −5.84535190561252109931500613042, −5.09243205839331280360160434984, −4.34946497982492537795616020356, −3.47811148791484563753097901740, −2.11647996569864938782797400389, −0.48995473117287849666172017208, 1.01885919840450273710011503831, 2.83420243268860622775016912094, 3.73524494010756263486615007086, 4.46059699682401092963907660475, 5.44072737716550895036323707954, 6.42302532798520175860641459080, 6.94242713079622930753056076414, 7.45280302120954850523857079122, 8.578760733048222878830945398542, 9.462106189837616005740731875146

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