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Properties

Label 2-768-24.11-c1-0-12
Degree $2$
Conductor $768$
Sign $0.408 - 0.912i$
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  + (0.618 + 1.61i)3-s + 1.23·5-s − 3.23i·7-s + (−2.23 + 2.00i)9-s + 0.763i·11-s + 4.47i·13-s + (0.763 + 2.00i)15-s + 6.47i·17-s + 5.23·19-s + (5.23 − 2.00i)21-s + 6.47·23-s − 3.47·25-s + (−4.61 − 2.38i)27-s + 9.23·29-s − 0.763i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + 0.552·5-s − 1.22i·7-s + (−0.745 + 0.666i)9-s + 0.230i·11-s + 1.24i·13-s + (0.197 + 0.516i)15-s + 1.56i·17-s + 1.20·19-s + (1.14 − 0.436i)21-s + 1.34·23-s − 0.694·25-s + (−0.888 − 0.458i)27-s + 1.71·29-s − 0.137i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56985 + 1.01762i\)
\(L(\frac12)\) \(\approx\) \(1.56985 + 1.01762i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 - 1.61i)T \)
good5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 + 3.23iT - 7T^{2} \)
11 \( 1 - 0.763iT - 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 - 6.47iT - 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 - 9.23T + 29T^{2} \)
31 \( 1 + 0.763iT - 31T^{2} \)
37 \( 1 - 0.472iT - 37T^{2} \)
41 \( 1 - 2.47iT - 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 - 3.23iT - 59T^{2} \)
61 \( 1 + 8.47iT - 61T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 13.7iT - 79T^{2} \)
83 \( 1 + 7.23iT - 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 8.47T + 97T^{2} \)
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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

−10.32318757223453145073239624383, −9.750953125497315800468477289560, −8.958271565234225082256350097080, −8.008908528838254671261920918595, −7.01588506882836258599405623548, −6.05194532522764385833944496954, −4.82030855904975558518792360855, −4.13399543037172595332335176138, −3.11232730619813553288894916702, −1.57529871934387436279978745277, 1.02081493097523189981406554888, 2.62124695530669829008843234732, 3.03795318017681575240638062702, 5.17121339150905459758234223314, 5.66507361989607478536249095400, 6.71368811642471717302431040962, 7.59798614506119529994196141514, 8.483212531289877669703548751814, 9.203463525811448549476932953378, 9.933578679137095816712990570155

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