Properties

Label 2-768-12.11-c3-0-84
Degree $2$
Conductor $768$
Sign $-0.881 + 0.471i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.58 − 2.44i)3-s − 17.2i·5-s + 0.269i·7-s + (15 − 22.4i)9-s + 41.1·11-s − 64.6·13-s + (−42.3 − 79.1i)15-s + 63.4i·17-s − 154. i·19-s + (0.660 + 1.23i)21-s + 43.3·23-s − 173.·25-s + (13.7 − 139. i)27-s − 240. i·29-s + 79.1i·31-s + ⋯
L(s)  = 1  + (0.881 − 0.471i)3-s − 1.54i·5-s + 0.0145i·7-s + (0.555 − 0.831i)9-s + 1.12·11-s − 1.37·13-s + (−0.728 − 1.36i)15-s + 0.904i·17-s − 1.85i·19-s + (0.00686 + 0.0128i)21-s + 0.392·23-s − 1.38·25-s + (0.0979 − 0.995i)27-s − 1.54i·29-s + 0.458i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.419369954\)
\(L(\frac12)\) \(\approx\) \(2.419369954\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.58 + 2.44i)T \)
good5 \( 1 + 17.2iT - 125T^{2} \)
7 \( 1 - 0.269iT - 343T^{2} \)
11 \( 1 - 41.1T + 1.33e3T^{2} \)
13 \( 1 + 64.6T + 2.19e3T^{2} \)
17 \( 1 - 63.4iT - 4.91e3T^{2} \)
19 \( 1 + 154. iT - 6.85e3T^{2} \)
23 \( 1 - 43.3T + 1.21e4T^{2} \)
29 \( 1 + 240. iT - 2.43e4T^{2} \)
31 \( 1 - 79.1iT - 2.97e4T^{2} \)
37 \( 1 - 132.T + 5.06e4T^{2} \)
41 \( 1 - 101. iT - 6.89e4T^{2} \)
43 \( 1 - 120. iT - 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 - 6.17iT - 1.48e5T^{2} \)
59 \( 1 + 45.8T + 2.05e5T^{2} \)
61 \( 1 + 651.T + 2.26e5T^{2} \)
67 \( 1 - 685. iT - 3.00e5T^{2} \)
71 \( 1 + 836.T + 3.57e5T^{2} \)
73 \( 1 + 285.T + 3.89e5T^{2} \)
79 \( 1 + 940. iT - 4.93e5T^{2} \)
83 \( 1 + 1.01e3T + 5.71e5T^{2} \)
89 \( 1 - 432. iT - 7.04e5T^{2} \)
97 \( 1 - 1.00e3T + 9.12e5T^{2} \)
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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

−9.191508056740377419602194735838, −8.918625370083633852819301466864, −7.966823866376493247901035432224, −7.12753946856285602396565284379, −6.12138820555734170518341523687, −4.76281912752652846075404484698, −4.21390535504336320921433460897, −2.75336475172500638270112066719, −1.58227249026353971826223754127, −0.56412099025881594472420908146, 1.82040045636014187829203214168, 2.90664326841989751315669595374, 3.58432429235747296709515203147, 4.67597953801396356894855009897, 5.99262089468109186013877575751, 7.19089886145235802969296671365, 7.41432625494081199029137397270, 8.683011512182906276628024894477, 9.635521575334396764842461021790, 10.10781654354621293035018558747

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