Properties

Degree $2$
Conductor $2304$
Sign $0.577 - 0.816i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s + 2.82i·7-s + 4·11-s + 2·13-s − 1.41i·17-s − 5.65i·19-s + 4·23-s + 2.99·25-s + 7.07i·29-s − 8.48i·31-s − 4.00·35-s + 8·37-s − 4.24i·41-s + 11.3i·43-s − 12·47-s + ⋯
L(s)  = 1  + 0.632i·5-s + 1.06i·7-s + 1.20·11-s + 0.554·13-s − 0.342i·17-s − 1.29i·19-s + 0.834·23-s + 0.599·25-s + 1.31i·29-s − 1.52i·31-s − 0.676·35-s + 1.31·37-s − 0.662i·41-s + 1.72i·43-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Motivic weight: \(1\)
Character: $\chi_{2304} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038506609\)
\(L(\frac12)\) \(\approx\) \(2.038506609\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 5.65iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + 2.82iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 15.5iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.182796358858131062005746422852, −8.539105003596820625964827664326, −7.49112417903938353981118073297, −6.65999770087998053024576615956, −6.19019991054668863312000503695, −5.18977828004671580578011363272, −4.31004380098127609361863092683, −3.16424180665631368706856515975, −2.52494194985834679533180471482, −1.16084756680104319818946279339, 0.876164686237853542763896459170, 1.66296134316450211293625620685, 3.34591889857637317195337084234, 4.01354161046819685313686401912, 4.74871547394549170034787114716, 5.81178760462294483788582839207, 6.62178011092317198411646866205, 7.26508398200434080060145066892, 8.352174433618556756146477667705, 8.671132159022916897196033894620

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