Properties

Label 2-768-8.5-c3-0-44
Degree $2$
Conductor $768$
Sign $-0.707 - 0.707i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 14i·5-s − 24·7-s − 9·9-s − 28i·11-s − 74i·13-s − 42·15-s + 82·17-s − 92i·19-s + 72i·21-s + 8·23-s − 71·25-s + 27i·27-s − 138i·29-s − 80·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.25i·5-s − 1.29·7-s − 0.333·9-s − 0.767i·11-s − 1.57i·13-s − 0.722·15-s + 1.16·17-s − 1.11i·19-s + 0.748i·21-s + 0.0725·23-s − 0.568·25-s + 0.192i·27-s − 0.883i·29-s − 0.463·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.029883447\)
\(L(\frac12)\) \(\approx\) \(1.029883447\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
good5 \( 1 + 14iT - 125T^{2} \)
7 \( 1 + 24T + 343T^{2} \)
11 \( 1 + 28iT - 1.33e3T^{2} \)
13 \( 1 + 74iT - 2.19e3T^{2} \)
17 \( 1 - 82T + 4.91e3T^{2} \)
19 \( 1 + 92iT - 6.85e3T^{2} \)
23 \( 1 - 8T + 1.21e4T^{2} \)
29 \( 1 + 138iT - 2.43e4T^{2} \)
31 \( 1 + 80T + 2.97e4T^{2} \)
37 \( 1 + 30iT - 5.06e4T^{2} \)
41 \( 1 + 282T + 6.89e4T^{2} \)
43 \( 1 - 4iT - 7.95e4T^{2} \)
47 \( 1 + 240T + 1.03e5T^{2} \)
53 \( 1 - 130iT - 1.48e5T^{2} \)
59 \( 1 - 596iT - 2.05e5T^{2} \)
61 \( 1 + 218iT - 2.26e5T^{2} \)
67 \( 1 - 436iT - 3.00e5T^{2} \)
71 \( 1 - 856T + 3.57e5T^{2} \)
73 \( 1 - 998T + 3.89e5T^{2} \)
79 \( 1 - 32T + 4.93e5T^{2} \)
83 \( 1 - 1.50e3iT - 5.71e5T^{2} \)
89 \( 1 - 246T + 7.04e5T^{2} \)
97 \( 1 - 866T + 9.12e5T^{2} \)
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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

−9.357980000937952687519437417531, −8.468018738034087619685244699889, −7.83562745865748635318416457442, −6.71365559187858951590043620734, −5.74551550750571376064039165237, −5.15055055744073820492781780992, −3.61171907099952536160179163117, −2.76678432207595766815678875213, −0.984356794191576457958615555067, −0.33227264278853253182183223493, 1.92615398707818483023844590106, 3.25465306421631206924755681275, 3.73665523670741735984778100129, 5.09051760507480844054882651891, 6.36488920915821683270876777282, 6.76412950236055971932347983173, 7.73507435079042067658638917298, 9.062189721558620557559479501246, 9.843096620261683063123788637639, 10.19163377475394451099004904742

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