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Properties

Label 2-768-3.2-c2-0-41
Degree $2$
Conductor $768$
Sign $0.942 + 0.333i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 + i)3-s + (7.00 + 5.65i)9-s − 14i·11-s − 33.9i·17-s + 16.9·19-s + 25·25-s + (14.1 + 23.0i)27-s + (14 − 39.5i)33-s + 67.8i·41-s + 84.8·43-s − 49·49-s + (33.9 − 96i)51-s + (48 + 16.9i)57-s − 82i·59-s + 118.·67-s + ⋯
L(s)  = 1  + (0.942 + 0.333i)3-s + (0.777 + 0.628i)9-s − 1.27i·11-s − 1.99i·17-s + 0.893·19-s + 25-s + (0.523 + 0.851i)27-s + (0.424 − 1.19i)33-s + 1.65i·41-s + 1.97·43-s − 0.999·49-s + (0.665 − 1.88i)51-s + (0.842 + 0.297i)57-s − 1.38i·59-s + 1.77·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.942 + 0.333i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.713505757\)
\(L(\frac12)\) \(\approx\) \(2.713505757\)
\(L(2)\) 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 \( 1 + (-2.82 - i)T \)
good5 \( 1 - 25T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 + 14iT - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 + 33.9iT - 289T^{2} \)
19 \( 1 - 16.9T + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 - 67.8iT - 1.68e3T^{2} \)
43 \( 1 - 84.8T + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + 82iT - 3.48e3T^{2} \)
61 \( 1 + 3.72e3T^{2} \)
67 \( 1 - 118.T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 142T + 5.32e3T^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 - 158iT - 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + 94T + 9.40e3T^{2} \)
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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

−9.780569734774225477245112981792, −9.292164242521792348664134428133, −8.427774916932308130465791825870, −7.63495168565673535699088155094, −6.76040856783115788878324961051, −5.45073797576647302856782005838, −4.58491949272348302023670203975, −3.28786986920726198447903194643, −2.70411194709296404025987565066, −0.950199002863151206450412548067, 1.36875101064176717597572361048, 2.39917421385043750282690161181, 3.64266858952533784618349501352, 4.49799617493307901531764614797, 5.83815802319003284027986191977, 6.94029189952588883552514372375, 7.57143696047222344454862934824, 8.478719742139602842851378062778, 9.225316569848696207496495115851, 10.07762701164365665912290449968

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