Properties

Degree $2$
Conductor $2304$
Sign $-0.382 - 0.923i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)5-s − 4.24i·7-s + (−2.82 + 2.82i)11-s + (3 + 3i)13-s + 6·17-s + (−1.41 − 1.41i)19-s − 2.82i·23-s − 3i·25-s + (−4 − 4i)29-s − 4.24·31-s + (8.48 + 8.48i)35-s + (3 − 3i)37-s + 10i·41-s + (−4.24 + 4.24i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (−0.894 + 0.894i)5-s − 1.60i·7-s + (−0.852 + 0.852i)11-s + (0.832 + 0.832i)13-s + 1.45·17-s + (−0.324 − 0.324i)19-s − 0.589i·23-s − 0.600i·25-s + (−0.742 − 0.742i)29-s − 0.762·31-s + (1.43 + 1.43i)35-s + (0.493 − 0.493i)37-s + 1.56i·41-s + (−0.646 + 0.646i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.382 - 0.923i$
Motivic weight: \(1\)
Character: $\chi_{2304} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8410156556\)
\(L(\frac12)\) \(\approx\) \(0.8410156556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (2 - 2i)T - 5iT^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (1.41 + 1.41i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (4 + 4i)T + 29iT^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + (4.24 - 4.24i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (4 - 4i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (-3 - 3i)T + 61iT^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.460782334612069539042251376314, −7.963375893695145091242750927877, −7.83047922738050491503937253599, −6.98269244286854033136201617732, −6.46787669523686087911171125593, −5.16784833704585551655689096009, −4.10765781821088163764500017615, −3.76489249449833151399478317322, −2.66509754649516254128699670644, −1.20732504073449967939306802557, 0.31732378595242006805976043831, 1.73884290339946242779963227575, 3.14274292787743549056078545070, 3.62421376643498998759795845427, 5.11198517483063446117853502887, 5.46575774805932600455139064870, 6.12037577768347044742358463764, 7.59648121307244308576720067545, 8.120962446745646834497444679672, 8.641717433604495954971281477584

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