Properties

Label 2-768-24.5-c2-0-9
Degree $2$
Conductor $768$
Sign $-0.229 - 0.973i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.55 − 1.57i)3-s + 1.31·5-s + 10.2·7-s + (4.01 + 8.05i)9-s − 16.6·11-s + 18.7i·13-s + (−3.35 − 2.07i)15-s − 4.38i·17-s + 11.5i·19-s + (−26.1 − 16.1i)21-s − 16.7i·23-s − 23.2·25-s + (2.46 − 26.8i)27-s − 12.5·29-s − 20.3·31-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)3-s + 0.263·5-s + 1.46·7-s + (0.446 + 0.894i)9-s − 1.51·11-s + 1.44i·13-s + (−0.223 − 0.138i)15-s − 0.257i·17-s + 0.608i·19-s + (−1.24 − 0.769i)21-s − 0.728i·23-s − 0.930·25-s + (0.0912 − 0.995i)27-s − 0.432·29-s − 0.655·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8151301374\)
\(L(\frac12)\) \(\approx\) \(0.8151301374\)
\(L(2)\) 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 \( 1 + (2.55 + 1.57i)T \)
good5 \( 1 - 1.31T + 25T^{2} \)
7 \( 1 - 10.2T + 49T^{2} \)
11 \( 1 + 16.6T + 121T^{2} \)
13 \( 1 - 18.7iT - 169T^{2} \)
17 \( 1 + 4.38iT - 289T^{2} \)
19 \( 1 - 11.5iT - 361T^{2} \)
23 \( 1 + 16.7iT - 529T^{2} \)
29 \( 1 + 12.5T + 841T^{2} \)
31 \( 1 + 20.3T + 961T^{2} \)
37 \( 1 - 18.5iT - 1.36e3T^{2} \)
41 \( 1 - 78.6iT - 1.68e3T^{2} \)
43 \( 1 + 36.4iT - 1.84e3T^{2} \)
47 \( 1 + 19.9iT - 2.20e3T^{2} \)
53 \( 1 + 81.3T + 2.80e3T^{2} \)
59 \( 1 + 29.9T + 3.48e3T^{2} \)
61 \( 1 - 72.0iT - 3.72e3T^{2} \)
67 \( 1 - 56.3iT - 4.48e3T^{2} \)
71 \( 1 - 136. iT - 5.04e3T^{2} \)
73 \( 1 - 80.8T + 5.32e3T^{2} \)
79 \( 1 + 86.0T + 6.24e3T^{2} \)
83 \( 1 - 80.4T + 6.88e3T^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 + 20.4T + 9.40e3T^{2} \)
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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

−10.57241758993383898728414227118, −9.687931418991290132497866096326, −8.360019573386589045375093978442, −7.79425799981871998268134803838, −6.92225257084007385721067853268, −5.83542232554378183909807283169, −5.06780610030721665614111238258, −4.33123994357510558014824390413, −2.32550387668540711405197787135, −1.49383370940795857436527369356, 0.30628457991941962999629063557, 1.90455902363982241280563760747, 3.39636458138754262286173112726, 4.77276519831034739102237650206, 5.30658510416939246234181376310, 5.96741622540663947195410646377, 7.55238746835891986332560972489, 7.925210829509042332370039822004, 9.150984098445991364097789955981, 10.11816131286708493484164957095

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