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Properties

Label 2-768-48.11-c1-0-24
Degree $2$
Conductor $768$
Sign $-0.533 + 0.845i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 + 1.70i)3-s + (−1.41 + 1.41i)5-s + 1.41·7-s + (−2.82 − i)9-s + (−4 − 4i)11-s + (−3 + 3i)13-s + (−1.99 − 2.82i)15-s − 2.82i·17-s + (−2.82 − 2.82i)19-s + (−0.414 + 2.41i)21-s + 0.999i·25-s + (2.53 − 4.53i)27-s + (7.07 + 7.07i)29-s − 4.24i·31-s + (8 − 5.65i)33-s + ⋯
L(s)  = 1  + (−0.169 + 0.985i)3-s + (−0.632 + 0.632i)5-s + 0.534·7-s + (−0.942 − 0.333i)9-s + (−1.20 − 1.20i)11-s + (−0.832 + 0.832i)13-s + (−0.516 − 0.730i)15-s − 0.685i·17-s + (−0.648 − 0.648i)19-s + (−0.0903 + 0.526i)21-s + 0.199i·25-s + (0.487 − 0.872i)27-s + (1.31 + 1.31i)29-s − 0.762i·31-s + (1.39 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.533 + 0.845i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 - 1.70i)T \)
good5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + (4 + 4i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (2.82 + 2.82i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-7.07 - 7.07i)T + 29iT^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + (-2 - 2i)T + 59iT^{2} \)
61 \( 1 + (9 - 9i)T - 61iT^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 + 4iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 9.89iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.26528283474407834100264182192, −9.152781165866375228456190565896, −8.434336463029184434927050798715, −7.50162262472410572006268760461, −6.52505020410662522884546442593, −5.26894931395168316009317067777, −4.68218365006410760724955439231, −3.45726253351082013175244620406, −2.60453991830468270085188868922, 0, 1.68528658842869710819435331171, 2.82098217772694073074319543441, 4.54412017543256674680123819193, 5.12976024824030426543434884612, 6.30279550894550304746199819764, 7.36890589992249971637652691220, 8.174413754844200914320553244592, 8.297069146161249310177982491754, 9.966585554455217106913315121941

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