Properties

Degree $2$
Conductor $2304$
Sign $-0.707 + 0.707i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·5-s − 4·7-s + 2i·11-s + 2i·13-s + 2·17-s − 2i·19-s − 4·23-s + 25-s + 6i·29-s − 8i·35-s − 10i·37-s − 6·41-s + 6i·43-s − 8·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.894i·5-s − 1.51·7-s + 0.603i·11-s + 0.554i·13-s + 0.485·17-s − 0.458i·19-s − 0.834·23-s + 0.200·25-s + 1.11i·29-s − 1.35i·35-s − 1.64i·37-s − 0.937·41-s + 0.914i·43-s − 1.16·47-s + 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.707 + 0.707i$
Motivic weight: \(1\)
Character: $\chi_{2304} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.901283822911148933219274961961, −7.78807401181537754550940713269, −6.90658693472445454577262700351, −6.63217600020591710133260327071, −5.75245531885267310765400026046, −4.65041292432824869473491905227, −3.54504386175201089083978945511, −3.02134894909850940868110394782, −1.89489278127510207413757070938, 0, 1.21146561810893222769175172833, 2.76949698107409344784867387252, 3.51512988593556574794619540593, 4.43840339046971541107811169446, 5.54545642580210773785881917936, 6.05413092799802394714867394531, 6.88483364047604802017468384205, 7.927597082960768217010357813251, 8.537883598187149394139660939031

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