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Properties

Label 2-48e2-144.67-c0-0-3
Degree $2$
Conductor $2304$
Sign $-0.953 + 0.300i$
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  + (−0.965 − 0.258i)3-s + (−0.366 − 1.36i)5-s + (−0.707 − 1.22i)7-s + (0.866 + 0.499i)9-s + (0.258 − 0.965i)11-s + 1.41i·15-s + 17-s + (0.707 − 0.707i)19-s + (0.366 + 1.36i)21-s + (−0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (0.366 − 1.36i)29-s + (−0.499 + 0.866i)33-s + (−1.41 + 1.41i)35-s + (−1 + i)37-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.366 − 1.36i)5-s + (−0.707 − 1.22i)7-s + (0.866 + 0.499i)9-s + (0.258 − 0.965i)11-s + 1.41i·15-s + 17-s + (0.707 − 0.707i)19-s + (0.366 + 1.36i)21-s + (−0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (0.366 − 1.36i)29-s + (−0.499 + 0.866i)33-s + (−1.41 + 1.41i)35-s + (−1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ -0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6612500389\)
\(L(\frac12)\) \(\approx\) \(0.6612500389\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
good5 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 + 2iT - T^{2} \)
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

−8.768448930994648710966615889390, −8.011067377124768621014718770687, −7.27216004711330503321539756481, −6.48798390166178623419576053464, −5.65583975000493913749754121238, −4.88857928000784218077219277490, −4.14045870266340480279604753044, −3.25355983149810943388192387045, −1.27512592687237625361779638423, −0.58444938326949772310230701514, 1.82457131599065628903174205485, 3.13653009476224720845955724354, 3.65352161979215552104520098143, 5.00710147469265392815892483693, 5.65693096708500639785170281440, 6.49447534062750375247719661114, 7.00487611827268991593338446766, 7.74888779798096049623335454626, 8.992922389122785110147053585792, 9.730897015257244895176002541509

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