Properties

Label 2-768-24.5-c2-0-38
Degree $2$
Conductor $768$
Sign $0.430 + 0.902i$
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 − 5.65·5-s − 6·7-s + (7.00 + 5.65i)9-s + 5.65·11-s − 10i·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s + 2i·19-s + (−16.9 − 6i)21-s − 11.3i·23-s + 7.00·25-s + (14.1 + 23.0i)27-s + 16.9·29-s + 22·31-s + ⋯
L(s)  = 1  + (0.942 + 0.333i)3-s − 1.13·5-s − 0.857·7-s + (0.777 + 0.628i)9-s + 0.514·11-s − 0.769i·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s + 0.105i·19-s + (−0.808 − 0.285i)21-s − 0.491i·23-s + 0.280·25-s + (0.523 + 0.851i)27-s + 0.585·29-s + 0.709·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.430 + 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.553127306\)
\(L(\frac12)\) \(\approx\) \(1.553127306\)
\(L(2)\) 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 \( 1 + 5.65T + 25T^{2} \)
7 \( 1 + 6T + 49T^{2} \)
11 \( 1 - 5.65T + 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 + 22.6iT - 289T^{2} \)
19 \( 1 - 2iT - 361T^{2} \)
23 \( 1 + 11.3iT - 529T^{2} \)
29 \( 1 - 16.9T + 841T^{2} \)
31 \( 1 - 22T + 961T^{2} \)
37 \( 1 + 6iT - 1.36e3T^{2} \)
41 \( 1 + 33.9iT - 1.68e3T^{2} \)
43 \( 1 + 82iT - 1.84e3T^{2} \)
47 \( 1 + 67.8iT - 2.20e3T^{2} \)
53 \( 1 - 62.2T + 2.80e3T^{2} \)
59 \( 1 - 73.5T + 3.48e3T^{2} \)
61 \( 1 - 86iT - 3.72e3T^{2} \)
67 \( 1 - 2iT - 4.48e3T^{2} \)
71 \( 1 + 124. iT - 5.04e3T^{2} \)
73 \( 1 + 82T + 5.32e3T^{2} \)
79 \( 1 + 10T + 6.24e3T^{2} \)
83 \( 1 - 73.5T + 6.88e3T^{2} \)
89 \( 1 - 33.9iT - 7.92e3T^{2} \)
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

−9.957858655489764382936684146663, −8.989786328070716029186672032782, −8.375601923955218015872293386517, −7.40941316809987609939453885207, −6.81672755673568507080726123486, −5.32130397745471443043517603631, −4.16682248739447683973938964032, −3.46215419941368444778660995551, −2.54518158186459026504426355315, −0.51522788665765764518583575168, 1.30717438850800961820420181827, 2.83047362663289718757459606306, 3.77560825867312129419565300719, 4.39128476772537536874481324836, 6.24563780059640552444061913941, 6.83607013004234900522574519182, 7.86744763162179087753224566702, 8.418056149604268284829290427589, 9.366948100735767315706380301949, 10.02171452916135042036353384032

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