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Properties

Label 2-768-8.5-c3-0-31
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 + 18i·5-s − 8·7-s − 9·9-s − 36i·11-s − 10i·13-s − 54·15-s + 18·17-s − 100i·19-s − 24i·21-s − 72·23-s − 199·25-s − 27i·27-s − 234i·29-s − 16·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.60i·5-s − 0.431·7-s − 0.333·9-s − 0.986i·11-s − 0.213i·13-s − 0.929·15-s + 0.256·17-s − 1.20i·19-s − 0.249i·21-s − 0.652·23-s − 1.59·25-s − 0.192i·27-s − 1.49i·29-s − 0.0926·31-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

\(L(2)\) \(\approx\) \(0.9911234374\)
\(L(\frac12)\) \(\approx\) \(0.9911234374\)
\(L(\frac{5}{2})\) 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 \( 1 - 18iT - 125T^{2} \)
7 \( 1 + 8T + 343T^{2} \)
11 \( 1 + 36iT - 1.33e3T^{2} \)
13 \( 1 + 10iT - 2.19e3T^{2} \)
17 \( 1 - 18T + 4.91e3T^{2} \)
19 \( 1 + 100iT - 6.85e3T^{2} \)
23 \( 1 + 72T + 1.21e4T^{2} \)
29 \( 1 + 234iT - 2.43e4T^{2} \)
31 \( 1 + 16T + 2.97e4T^{2} \)
37 \( 1 - 226iT - 5.06e4T^{2} \)
41 \( 1 + 90T + 6.89e4T^{2} \)
43 \( 1 + 452iT - 7.95e4T^{2} \)
47 \( 1 - 432T + 1.03e5T^{2} \)
53 \( 1 + 414iT - 1.48e5T^{2} \)
59 \( 1 - 684iT - 2.05e5T^{2} \)
61 \( 1 - 422iT - 2.26e5T^{2} \)
67 \( 1 - 332iT - 3.00e5T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 + 26T + 3.89e5T^{2} \)
79 \( 1 - 512T + 4.93e5T^{2} \)
83 \( 1 + 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 - 630T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−10.08881067324306421500029197481, −9.095109911960597086915410145036, −8.097568209121498821018862034662, −7.09377638524978727077694808192, −6.31591941622518621019637956563, −5.53549213916671942318093155972, −4.08148615729278025458022051626, −3.20113523901968485947521524927, −2.47795874944550601029461285231, −0.29027593025749778408846499592, 1.11729544705206604353791469849, 1.99966830368175073866053642340, 3.65672284550610119733189711989, 4.70651228336037168848334717424, 5.53859165242114754583245503607, 6.51312737586089496700968102969, 7.62279060452625006954183538401, 8.255384401163519501133556230548, 9.238811468744759972913469501715, 9.737884546594726063377782717581

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