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Properties

Label 2-48e2-12.11-c1-0-8
Degree $2$
Conductor $2304$
Sign $0.577 - 0.816i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
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: $0.577 - 0.816i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(1.463246573\)
\(L(\frac12)\) \(\approx\) \(1.463246573\)
\(L(\frac{3}{2})\) 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

−9.054391760774999360260587449735, −8.271091592999289937047171871556, −7.41735531102962315051526068878, −7.04329114412584343484852007096, −5.98180406864672585525847776756, −5.24159772156522462403328546607, −4.13883471506571946281117468663, −3.41797719529731272497047104348, −2.43209672067615054521389300493, −1.06425319391936915613189201724, 0.58277542301415223049391014972, 2.18220387070786472842303972083, 2.81830461708124090825308990807, 4.19799100720704853230427825494, 4.92407014084292868688175349228, 5.80247300573540949560544946006, 6.28208328102252349530996046477, 7.61293359811443231651344440672, 8.155284680866821212411763501806, 8.854265566896114308633366959102

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