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Properties

Label 2-768-8.5-c1-0-8
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 + 2·7-s − 9-s + 4i·11-s − 6i·13-s + 6·17-s − 2i·21-s + 4·23-s + 5·25-s + i·27-s − 4i·29-s − 10·31-s + 4·33-s + 2i·37-s − 6·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.755·7-s − 0.333·9-s + 1.20i·11-s − 1.66i·13-s + 1.45·17-s − 0.436i·21-s + 0.834·23-s + 25-s + 0.192i·27-s − 0.742i·29-s − 1.79·31-s + 0.696·33-s + 0.328i·37-s − 0.960·39-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

\(L(1)\) \(\approx\) \(1.56701 - 0.649076i\)
\(L(\frac12)\) \(\approx\) \(1.56701 - 0.649076i\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.32746818568192312839748502723, −9.368856051707355902230072640025, −8.275147052231148820384204512104, −7.62636888582029667729632876830, −6.98167379803965498680802887920, −5.56960999106934779616024988406, −5.08102881373563885577068223802, −3.61326847361635129245552726008, −2.38832576738429158256579444906, −1.05358884779538257442246104043, 1.36953664321006418648610239834, 2.99890171503348173611991854582, 4.01758698833620399433183443014, 5.03844313933873952201198379022, 5.83696971187077453581934174752, 6.99709311734244714856378557170, 7.937663423525615724717844559466, 8.980194246803884730916866604967, 9.305349933171789530809623443936, 10.71952455842196254021447580282

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