Properties

Label 2-768-12.11-c1-0-3
Degree $2$
Conductor $768$
Sign $-0.577 - 0.816i$
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  + (1 + 1.41i)3-s + (−1.00 + 2.82i)9-s − 6·11-s + 5.65i·17-s + 8.48i·19-s + 5·25-s + (−5.00 + 1.41i)27-s + (−6 − 8.48i)33-s − 11.3i·41-s + 8.48i·43-s + 7·49-s + (−8.00 + 5.65i)51-s + (−12 + 8.48i)57-s + 6·59-s − 8.48i·67-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (−0.333 + 0.942i)9-s − 1.80·11-s + 1.37i·17-s + 1.94i·19-s + 25-s + (−0.962 + 0.272i)27-s + (−1.04 − 1.47i)33-s − 1.76i·41-s + 1.29i·43-s + 49-s + (−1.12 + 0.792i)51-s + (−1.58 + 1.12i)57-s + 0.781·59-s − 1.03i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622150 + 1.20190i\)
\(L(\frac12)\) \(\approx\) \(0.622150 + 1.20190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 8.48iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 8.48iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
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

−10.34978052077095511992188404187, −10.08468491543284455159461073244, −8.790302036599455494425237302694, −8.162167699761703250175291054002, −7.49955535214414209473593430205, −5.95630682999826388559510429731, −5.23749201718865987480865849146, −4.13726076763144534312633150309, −3.19057485211952431031773564650, −2.04263476431434176938798796144, 0.60959836316426054745840675418, 2.49289565774882518970617759263, 2.96179654937690131647998420245, 4.69186723337989744382247992323, 5.52234145276171359986549740330, 6.89652540821914029109718029367, 7.31850640523112292206964249084, 8.301570930389600637753947718860, 9.014021864945821662185383300290, 9.926124375736567438467452817323

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