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Properties

Label 2-768-96.35-c1-0-3
Degree $2$
Conductor $768$
Sign $-0.806 - 0.591i$
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  + (1.68 + 0.406i)3-s + (−2.81 + 1.16i)5-s + (−0.543 − 0.543i)7-s + (2.66 + 1.36i)9-s + (−3.96 + 1.64i)11-s + (−1.13 + 2.74i)13-s + (−5.21 + 0.819i)15-s − 5.73·17-s + (−0.0793 − 0.0328i)19-s + (−0.694 − 1.13i)21-s + (1.46 + 1.46i)23-s + (3.02 − 3.02i)25-s + (3.93 + 3.38i)27-s + (0.520 − 1.25i)29-s + 5.64i·31-s + ⋯
L(s)  = 1  + (0.972 + 0.234i)3-s + (−1.25 + 0.521i)5-s + (−0.205 − 0.205i)7-s + (0.889 + 0.456i)9-s + (−1.19 + 0.495i)11-s + (−0.315 + 0.762i)13-s + (−1.34 + 0.211i)15-s − 1.39·17-s + (−0.0181 − 0.00753i)19-s + (−0.151 − 0.248i)21-s + (0.305 + 0.305i)23-s + (0.605 − 0.605i)25-s + (0.758 + 0.652i)27-s + (0.0966 − 0.233i)29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.806 - 0.591i$
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: \((2,\ 768,\ (\ :1/2),\ -0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270964 + 0.826885i\)
\(L(\frac12)\) \(\approx\) \(0.270964 + 0.826885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.406i)T \)
good5 \( 1 + (2.81 - 1.16i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.543 + 0.543i)T + 7iT^{2} \)
11 \( 1 + (3.96 - 1.64i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.13 - 2.74i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (0.0793 + 0.0328i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.46 - 1.46i)T + 23iT^{2} \)
29 \( 1 + (-0.520 + 1.25i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 5.64iT - 31T^{2} \)
37 \( 1 + (4.19 + 10.1i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.93 - 4.93i)T - 41iT^{2} \)
43 \( 1 + (-3.50 - 8.46i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.98iT - 47T^{2} \)
53 \( 1 + (-1.04 - 2.53i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.498 + 1.20i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.69 - 1.52i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.35 + 8.10i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.08 + 4.08i)T - 71iT^{2} \)
73 \( 1 + (-1.59 - 1.59i)T + 73iT^{2} \)
79 \( 1 - 0.637T + 79T^{2} \)
83 \( 1 + (1.94 - 4.69i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.902 + 0.902i)T + 89iT^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.71920663760707696707523599914, −9.751991221212580973434848707145, −8.897061585755475161071896936431, −8.012167259470330814408257719957, −7.34951170100188888867130231401, −6.71424507212936514122972672151, −4.90376638845658718131062555299, −4.17336905967443168199363515543, −3.19935298779353595802435898885, −2.19640213673607457867760172268, 0.36720816207623334519635975539, 2.36825900172326848293195080562, 3.34517796210375173270374905491, 4.33467433306666089291921689704, 5.32840627970300828891752989412, 6.79501904510270636095076465720, 7.59234314984764504273245161228, 8.442577069893497678296691928034, 8.669781125820495150919198079418, 9.941807685173233000601250493178

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