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Properties

Label 2-768-32.29-c1-0-11
Degree $2$
Conductor $768$
Sign $0.639 + 0.768i$
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 + (−0.750 − 1.81i)5-s + (−0.638 + 0.638i)7-s + (0.707 + 0.707i)9-s + (−0.343 + 0.142i)11-s + (1.56 − 3.78i)13-s − 1.96i·15-s − 1.52i·17-s + (3.15 − 7.61i)19-s + (−0.834 + 0.345i)21-s + (6.00 + 6.00i)23-s + (0.813 − 0.813i)25-s + (0.382 + 0.923i)27-s + (0.647 + 0.268i)29-s + 3.66·31-s + ⋯
L(s)  = 1  + (0.533 + 0.220i)3-s + (−0.335 − 0.810i)5-s + (−0.241 + 0.241i)7-s + (0.235 + 0.235i)9-s + (−0.103 + 0.0428i)11-s + (0.435 − 1.05i)13-s − 0.506i·15-s − 0.369i·17-s + (0.723 − 1.74i)19-s + (−0.182 + 0.0754i)21-s + (1.25 + 1.25i)23-s + (0.162 − 0.162i)25-s + (0.0736 + 0.177i)27-s + (0.120 + 0.0497i)29-s + 0.658·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.639 + 0.768i$
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: \((2,\ 768,\ (\ :1/2),\ 0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51816 - 0.711966i\)
\(L(\frac12)\) \(\approx\) \(1.51816 - 0.711966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.750 + 1.81i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.638 - 0.638i)T - 7iT^{2} \)
11 \( 1 + (0.343 - 0.142i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.56 + 3.78i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.52iT - 17T^{2} \)
19 \( 1 + (-3.15 + 7.61i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.00 - 6.00i)T + 23iT^{2} \)
29 \( 1 + (-0.647 - 0.268i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + (3.69 + 8.90i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.19 + 8.19i)T + 41iT^{2} \)
43 \( 1 + (1.86 - 0.771i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 3.21iT - 47T^{2} \)
53 \( 1 + (7.71 - 3.19i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.78 - 6.72i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-10.4 - 4.34i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-6.56 - 2.72i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (0.957 - 0.957i)T - 71iT^{2} \)
73 \( 1 + (2.14 + 2.14i)T + 73iT^{2} \)
79 \( 1 + 0.628iT - 79T^{2} \)
83 \( 1 + (4.17 - 10.0i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.70 + 8.70i)T - 89iT^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−10.09589817159766186506657389690, −9.057913942418871854370145870723, −8.779281512117066515383002769610, −7.68570577993676307304779092041, −6.93193751945325322849187945728, −5.44231263021668681958657480529, −4.89207870113175315653552815066, −3.59069226101770835657478900003, −2.68925165551433168610501644558, −0.883638383492771344781674961756, 1.55268573882468662473770855128, 3.02574386940791850254700331544, 3.71598353375511229345729935443, 4.94283696098442909066103834516, 6.51535586175050786748162790473, 6.76398150793906271816097015517, 7.974769859902580104110895941684, 8.546462277774568919901829092044, 9.709575536223308020879677493573, 10.33430989073977159083938315822

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