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Properties

Label 2-768-32.29-c1-0-3
Degree $2$
Conductor $768$
Sign $-0.172 - 0.985i$
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  + (0.923 + 0.382i)3-s + (0.155 + 0.375i)5-s + (−0.709 + 0.709i)7-s + (0.707 + 0.707i)9-s + (−2.79 + 1.15i)11-s + (−2.58 + 6.24i)13-s + 0.406i·15-s + 1.05i·17-s + (−1.48 + 3.59i)19-s + (−0.926 + 0.383i)21-s + (−0.922 − 0.922i)23-s + (3.41 − 3.41i)25-s + (0.382 + 0.923i)27-s + (7.64 + 3.16i)29-s + 1.88·31-s + ⋯
L(s)  = 1  + (0.533 + 0.220i)3-s + (0.0696 + 0.168i)5-s + (−0.268 + 0.268i)7-s + (0.235 + 0.235i)9-s + (−0.841 + 0.348i)11-s + (−0.717 + 1.73i)13-s + 0.105i·15-s + 0.256i·17-s + (−0.341 + 0.823i)19-s + (−0.202 + 0.0837i)21-s + (−0.192 − 0.192i)23-s + (0.683 − 0.683i)25-s + (0.0736 + 0.177i)27-s + (1.41 + 0.587i)29-s + 0.338·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.172 - 0.985i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.172 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917790 + 1.09200i\)
\(L(\frac12)\) \(\approx\) \(0.917790 + 1.09200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 - 0.382i)T \)
good5 \( 1 + (-0.155 - 0.375i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.709 - 0.709i)T - 7iT^{2} \)
11 \( 1 + (2.79 - 1.15i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.58 - 6.24i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 1.05iT - 17T^{2} \)
19 \( 1 + (1.48 - 3.59i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.922 + 0.922i)T + 23iT^{2} \)
29 \( 1 + (-7.64 - 3.16i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.88T + 31T^{2} \)
37 \( 1 + (1.24 + 3.01i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.11 - 5.11i)T + 41iT^{2} \)
43 \( 1 + (10.9 - 4.53i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 7.47iT - 47T^{2} \)
53 \( 1 + (-7.58 + 3.14i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.13 + 9.98i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.35 + 0.562i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-10.8 - 4.50i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.35 - 9.35i)T - 71iT^{2} \)
73 \( 1 + (-0.367 - 0.367i)T + 73iT^{2} \)
79 \( 1 - 5.87iT - 79T^{2} \)
83 \( 1 + (-1.62 + 3.91i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.33 + 8.33i)T - 89iT^{2} \)
97 \( 1 + 7.97T + 97T^{2} \)
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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

−10.23600779476410439601340867847, −9.860609081475088060715677492943, −8.808412571895803881358585417282, −8.152539396452264308137183636010, −7.04280035699630804857451535931, −6.34653669753111365004188608430, −4.98375711637489762236378924653, −4.21549809828034098544248381954, −2.89408174646919589499747898935, −1.95447351306260199915846898498, 0.65703066019578502544790742126, 2.55395490345377611715523664398, 3.24560454725874290633434985634, 4.71959740574546235562970124502, 5.54269897772524553261324435574, 6.73154078182927060137849103037, 7.63876663956504226905461868170, 8.277546982073915033862231539824, 9.174420643707457461444606060017, 10.23483534736330571751064176155

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