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Properties

Label 2-768-8.5-c1-0-3
Degree $2$
Conductor $768$
Sign $-0.707 - 0.707i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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  + i·3-s + 4i·5-s + 2·7-s − 9-s + 4i·11-s + 2i·13-s − 4·15-s − 2·17-s − 8i·19-s + 2i·21-s + 4·23-s − 11·25-s − i·27-s + 6·31-s − 4·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.78i·5-s + 0.755·7-s − 0.333·9-s + 1.20i·11-s + 0.554i·13-s − 1.03·15-s − 0.485·17-s − 1.83i·19-s + 0.436i·21-s + 0.834·23-s − 2.20·25-s − 0.192i·27-s + 1.07·31-s − 0.696·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567863 + 1.37094i\)
\(L(\frac12)\) \(\approx\) \(0.567863 + 1.37094i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 14iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.70206185831713788930245676267, −9.916600997518144663362488225329, −9.135333050200086696728605841255, −7.976423808355417881524759290288, −6.89746921824129312899974225309, −6.66507801682956302120994236992, −5.06817891142303539060681617542, −4.32986925465745354573808228271, −3.03471852115787710647152141334, −2.16416084766877351097921162649, 0.77858141589193121347441626816, 1.76585453833573443385017399365, 3.50353875069208416710026970920, 4.75363724414827720741509638280, 5.46098441734282868973399766682, 6.30042210312539327070286376552, 7.83814253456007657478137667272, 8.278326865716987156084613925871, 8.821521414159894913227301420852, 9.896673563496253942721087298003

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