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Properties

Label 2-768-96.35-c1-0-16
Degree $2$
Conductor $768$
Sign $0.997 + 0.0677i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 − 0.722i)3-s + (0.378 − 0.156i)5-s + (2.01 + 2.01i)7-s + (1.95 − 2.27i)9-s + (0.709 − 0.294i)11-s + (−2.08 + 5.03i)13-s + (0.482 − 0.520i)15-s + 6.33·17-s + (−0.646 − 0.267i)19-s + (4.61 + 1.71i)21-s + (−1.61 − 1.61i)23-s + (−3.41 + 3.41i)25-s + (1.43 − 4.99i)27-s + (2.04 − 4.93i)29-s − 5.75i·31-s + ⋯
L(s)  = 1  + (0.908 − 0.417i)3-s + (0.169 − 0.0701i)5-s + (0.760 + 0.760i)7-s + (0.651 − 0.758i)9-s + (0.214 − 0.0886i)11-s + (−0.577 + 1.39i)13-s + (0.124 − 0.134i)15-s + 1.53·17-s + (−0.148 − 0.0614i)19-s + (1.00 + 0.373i)21-s + (−0.337 − 0.337i)23-s + (−0.683 + 0.683i)25-s + (0.276 − 0.961i)27-s + (0.379 − 0.917i)29-s − 1.03i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.997 + 0.0677i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.997 + 0.0677i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36533 - 0.0801961i\)
\(L(\frac12)\) \(\approx\) \(2.36533 - 0.0801961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.722i)T \)
good5 \( 1 + (-0.378 + 0.156i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.01 - 2.01i)T + 7iT^{2} \)
11 \( 1 + (-0.709 + 0.294i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.08 - 5.03i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.33T + 17T^{2} \)
19 \( 1 + (0.646 + 0.267i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.61 + 1.61i)T + 23iT^{2} \)
29 \( 1 + (-2.04 + 4.93i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 5.75iT - 31T^{2} \)
37 \( 1 + (-2.50 - 6.04i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.52 - 5.52i)T - 41iT^{2} \)
43 \( 1 + (-0.406 - 0.981i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + (-0.674 - 1.62i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.35 + 8.08i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.14 + 1.71i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-2.65 + 6.40i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.97 + 1.97i)T - 71iT^{2} \)
73 \( 1 + (-9.48 - 9.48i)T + 73iT^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 + (6.60 - 15.9i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.11 + 6.11i)T + 89iT^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−9.831073560638446444305427499979, −9.572268108332905209730772920403, −8.435234508021424474670806357116, −7.965927735265773703497439758951, −6.94670877675530398744504659937, −5.98307347972732896931772512652, −4.82879804486633591306720514979, −3.75306749796140039396144446736, −2.44853672010099392431425026365, −1.57043925857173634348470707662, 1.38541759166282033742600583105, 2.83211747995069262108253834035, 3.77414445690950963658125846029, 4.82116095503026984804183637224, 5.70388488351735686594035629268, 7.28227133233092368572236500431, 7.76809329418090966651922847363, 8.509322737477710407462636670828, 9.609289282096183065413984649567, 10.33346183670509985070846038568

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