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Properties

Label 2-768-3.2-c2-0-49
Degree $2$
Conductor $768$
Sign $-0.666 + 0.745i$
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 − 2.23i)3-s + 4i·5-s − 8.94·7-s + (−1.00 − 8.94i)9-s − 4.47i·11-s + 17.8·13-s + (8.94 + 8i)15-s − 17.8i·17-s − 20·19-s + (−17.8 + 20.0i)21-s + 16i·23-s + 9·25-s + (−22.0 − 15.6i)27-s − 52i·29-s − 26.8·31-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)3-s + 0.800i·5-s − 1.27·7-s + (−0.111 − 0.993i)9-s − 0.406i·11-s + 1.37·13-s + (0.596 + 0.533i)15-s − 1.05i·17-s − 1.05·19-s + (−0.851 + 0.952i)21-s + 0.695i·23-s + 0.359·25-s + (−0.814 − 0.579i)27-s − 1.79i·29-s − 0.865·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.666 + 0.745i$
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: \((2,\ 768,\ (\ :1),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.277492220\)
\(L(\frac12)\) \(\approx\) \(1.277492220\)
\(L(2)\) 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 \( 1 + (-2 + 2.23i)T \)
good5 \( 1 - 4iT - 25T^{2} \)
7 \( 1 + 8.94T + 49T^{2} \)
11 \( 1 + 4.47iT - 121T^{2} \)
13 \( 1 - 17.8T + 169T^{2} \)
17 \( 1 + 17.8iT - 289T^{2} \)
19 \( 1 + 20T + 361T^{2} \)
23 \( 1 - 16iT - 529T^{2} \)
29 \( 1 + 52iT - 841T^{2} \)
31 \( 1 + 26.8T + 961T^{2} \)
37 \( 1 + 53.6T + 1.36e3T^{2} \)
41 \( 1 + 35.7iT - 1.68e3T^{2} \)
43 \( 1 + 36T + 1.84e3T^{2} \)
47 \( 1 + 64iT - 2.20e3T^{2} \)
53 \( 1 - 20iT - 2.80e3T^{2} \)
59 \( 1 + 102. iT - 3.48e3T^{2} \)
61 \( 1 - 17.8T + 3.72e3T^{2} \)
67 \( 1 + 44T + 4.48e3T^{2} \)
71 \( 1 - 80iT - 5.04e3T^{2} \)
73 \( 1 - 50T + 5.32e3T^{2} \)
79 \( 1 - 80.4T + 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 - 160. iT - 7.92e3T^{2} \)
97 \( 1 - 50T + 9.40e3T^{2} \)
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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

−9.686360342362426487152943754255, −8.923208523687528338481276832373, −8.123691181287954817794636783696, −6.95776691417840521613779656667, −6.58707751752186665155664976110, −5.69852322831963765490257175166, −3.74512100046242157105018741524, −3.23740409400839883690108703013, −2.10765152338605495486233640089, −0.38781954531104014354274751066, 1.62023312677747646096731307270, 3.15558779846777955086525971210, 3.90106535502129856597394102978, 4.87063134985177048097897712322, 6.01333714643961216195995043768, 6.88790177069895196943270257360, 8.299385815396771177564719336001, 8.779719967446302176218455447036, 9.397794551090893375740933539756, 10.51096789722698517631362910994

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