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Properties

Label 2-768-96.59-c1-0-10
Degree $2$
Conductor $768$
Sign $0.759 - 0.650i$
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  + (−1.69 + 0.361i)3-s + (0.348 − 0.842i)5-s + (−0.471 + 0.471i)7-s + (2.73 − 1.22i)9-s + (−1.24 + 2.99i)11-s + (2.51 − 1.04i)13-s + (−0.286 + 1.55i)15-s − 6.24·17-s + (0.683 + 1.65i)19-s + (0.628 − 0.969i)21-s + (5.69 − 5.69i)23-s + (2.94 + 2.94i)25-s + (−4.19 + 3.06i)27-s + (5.28 − 2.19i)29-s + 5.07i·31-s + ⋯
L(s)  = 1  + (−0.977 + 0.208i)3-s + (0.156 − 0.376i)5-s + (−0.178 + 0.178i)7-s + (0.912 − 0.408i)9-s + (−0.374 + 0.903i)11-s + (0.698 − 0.289i)13-s + (−0.0740 + 0.401i)15-s − 1.51·17-s + (0.156 + 0.378i)19-s + (0.137 − 0.211i)21-s + (1.18 − 1.18i)23-s + (0.589 + 0.589i)25-s + (−0.807 + 0.589i)27-s + (0.982 − 0.406i)29-s + 0.911i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991301 + 0.366486i\)
\(L(\frac12)\) \(\approx\) \(0.991301 + 0.366486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.361i)T \)
good5 \( 1 + (-0.348 + 0.842i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.471 - 0.471i)T - 7iT^{2} \)
11 \( 1 + (1.24 - 2.99i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.51 + 1.04i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.24T + 17T^{2} \)
19 \( 1 + (-0.683 - 1.65i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.69 + 5.69i)T - 23iT^{2} \)
29 \( 1 + (-5.28 + 2.19i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 5.07iT - 31T^{2} \)
37 \( 1 + (-6.21 - 2.57i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.43 - 6.43i)T + 41iT^{2} \)
43 \( 1 + (-6.04 - 2.50i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 + (4.63 + 1.91i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.67 + 1.10i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.36 - 5.70i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.39 + 1.40i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.71 + 7.71i)T + 71iT^{2} \)
73 \( 1 + (-0.0492 + 0.0492i)T - 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + (-3.65 + 1.51i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.09 + 4.09i)T - 89iT^{2} \)
97 \( 1 - 4.72T + 97T^{2} \)
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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

−10.68992276953437539726205064008, −9.564569961883700524346898111978, −8.931591759399149958098643394328, −7.76969996457394136013616699786, −6.66470216154860382130130326127, −6.10715693133152833530110743476, −4.84443257403633223344296390287, −4.45097234261586125018917039257, −2.77398205088310625645350164523, −1.13074558361017392796575596822, 0.75428146710495071190036166034, 2.43884207511415948533003848337, 3.84201243350755830699916126099, 4.94442944337149213463030026824, 5.92251632159001062414949017675, 6.63524767836790268095247844123, 7.37086481330545439491925241773, 8.578342065862741094939827015550, 9.412553828245253640069425542666, 10.59298712934985104699584011457

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