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Properties

Label 2-768-16.11-c2-0-1
Degree $2$
Conductor $768$
Sign $-0.382 - 0.923i$
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  + (−1.22 − 1.22i)3-s + (−2.73 − 2.73i)5-s − 2.17·7-s + 2.99i·9-s + (−0.277 + 0.277i)11-s + (13.3 − 13.3i)13-s + 6.69i·15-s − 26.3·17-s + (−2.17 − 2.17i)19-s + (2.66 + 2.66i)21-s − 7.52·23-s − 10.0i·25-s + (3.67 − 3.67i)27-s + (6.19 − 6.19i)29-s + 28.7i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.546 − 0.546i)5-s − 0.310·7-s + 0.333i·9-s + (−0.0252 + 0.0252i)11-s + (1.03 − 1.03i)13-s + 0.446i·15-s − 1.55·17-s + (−0.114 − 0.114i)19-s + (0.126 + 0.126i)21-s − 0.327·23-s − 0.402i·25-s + (0.136 − 0.136i)27-s + (0.213 − 0.213i)29-s + 0.927i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1850439974\)
\(L(\frac12)\) \(\approx\) \(0.1850439974\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (2.73 + 2.73i)T + 25iT^{2} \)
7 \( 1 + 2.17T + 49T^{2} \)
11 \( 1 + (0.277 - 0.277i)T - 121iT^{2} \)
13 \( 1 + (-13.3 + 13.3i)T - 169iT^{2} \)
17 \( 1 + 26.3T + 289T^{2} \)
19 \( 1 + (2.17 + 2.17i)T + 361iT^{2} \)
23 \( 1 + 7.52T + 529T^{2} \)
29 \( 1 + (-6.19 + 6.19i)T - 841iT^{2} \)
31 \( 1 - 28.7iT - 961T^{2} \)
37 \( 1 + (4.07 + 4.07i)T + 1.36e3iT^{2} \)
41 \( 1 - 23.1iT - 1.68e3T^{2} \)
43 \( 1 + (49.0 - 49.0i)T - 1.84e3iT^{2} \)
47 \( 1 - 90.4iT - 2.20e3T^{2} \)
53 \( 1 + (-37.5 - 37.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (44.8 - 44.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-57.4 + 57.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-57.8 - 57.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 46.5T + 5.04e3T^{2} \)
73 \( 1 - 2.43iT - 5.32e3T^{2} \)
79 \( 1 - 1.76iT - 6.24e3T^{2} \)
83 \( 1 + (58.5 + 58.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 - 46.9T + 9.40e3T^{2} \)
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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

−10.60845362036977545939809496669, −9.485522245590258452617971697020, −8.444846818550215972983847911522, −8.016251597198679890390543560513, −6.75824615407196170028925980676, −6.12326616452998502834623200852, −4.98398793580446387905333888052, −4.07988336112657051810711447757, −2.79606987243032042869852657878, −1.20809966186151305265741652820, 0.07275057786321122054214768751, 2.02279597592656698671223499601, 3.55812606687354954077223140312, 4.16332704317677913962150621585, 5.36957025369901526017356567718, 6.57345140237937472684223757744, 6.89908129194883902998902419275, 8.292562791211451921144144072652, 9.002363113538835117624771949798, 9.937532211930336194434456468162

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