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Properties

Label 2-768-12.11-c1-0-13
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

\(L(1)\) \(\approx\) \(1.20190 - 0.622150i\)
\(L(\frac12)\) \(\approx\) \(1.20190 - 0.622150i\)
\(L(\frac{3}{2})\) 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

−10.46033345850192554351588473598, −9.049635266734728437059057907380, −8.663013357009031343385744120378, −7.29678260320644648977236916626, −6.73788536351109334688842912040, −5.97665725329863433223916161740, −4.85296615641386915626758451398, −3.75030586513910626996503822543, −2.19796685980815850566575176915, −0.940193319874360338425704127048, 1.21325177578667372153513972808, 3.17748289583813502672208186173, 4.10097693297599374182001353842, 4.97038504677694939157559048920, 6.09965473708627903733411870985, 6.68738746488112880918032304464, 7.933233395447223951528066045261, 9.114235929380825439842956604773, 9.529679604819364368860346342313, 10.41028537391621499623050381498

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