Properties

Label 2-768-32.21-c1-0-6
Degree $2$
Conductor $768$
Sign $0.928 - 0.371i$
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 + (0.00259 − 0.00626i)5-s + (2.41 + 2.41i)7-s + (0.707 − 0.707i)9-s + (1.29 + 0.538i)11-s + (0.559 + 1.35i)13-s − 0.00678i·15-s + 5.82i·17-s + (−2.67 − 6.46i)19-s + (3.16 + 1.30i)21-s + (−0.178 + 0.178i)23-s + (3.53 + 3.53i)25-s + (0.382 − 0.923i)27-s + (−5.72 + 2.37i)29-s + 6.19·31-s + ⋯
L(s)  = 1  + (0.533 − 0.220i)3-s + (0.00116 − 0.00280i)5-s + (0.914 + 0.914i)7-s + (0.235 − 0.235i)9-s + (0.391 + 0.162i)11-s + (0.155 + 0.374i)13-s − 0.00175i·15-s + 1.41i·17-s + (−0.614 − 1.48i)19-s + (0.689 + 0.285i)21-s + (−0.0372 + 0.0372i)23-s + (0.707 + 0.707i)25-s + (0.0736 − 0.177i)27-s + (−1.06 + 0.440i)29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02046 + 0.389004i\)
\(L(\frac12)\) \(\approx\) \(2.02046 + 0.389004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.00259 + 0.00626i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 + (-1.29 - 0.538i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.559 - 1.35i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.82iT - 17T^{2} \)
19 \( 1 + (2.67 + 6.46i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.178 - 0.178i)T - 23iT^{2} \)
29 \( 1 + (5.72 - 2.37i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 + (-2.02 + 4.89i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.36 - 3.36i)T - 41iT^{2} \)
43 \( 1 + (-9.37 - 3.88i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.5iT - 47T^{2} \)
53 \( 1 + (-8.36 - 3.46i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.59 + 3.85i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (7.27 - 3.01i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (4.38 - 1.81i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.95 + 5.95i)T + 71iT^{2} \)
73 \( 1 + (-7.85 + 7.85i)T - 73iT^{2} \)
79 \( 1 - 1.42iT - 79T^{2} \)
83 \( 1 + (3.03 + 7.32i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.96 + 9.96i)T + 89iT^{2} \)
97 \( 1 + 1.24T + 97T^{2} \)
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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

−10.45625645806283593601274832384, −9.071509310916741148611423088002, −8.866152648146228443410329998728, −7.927620138067697079526689412921, −6.95543457815817706791514594090, −6.00566807839395145947654858323, −4.91990336402558996757554157500, −3.93625565837600041751936642750, −2.55687678072592569330783827634, −1.58272708094943284569306849108, 1.16384964590680168430444980283, 2.63130514911502570473515608837, 3.92506446377423153381291182123, 4.60540670811406105760445955645, 5.79033651810181583882132142271, 6.98515176450387188011234716776, 7.82588483453212822651039663834, 8.426042826194131478867694712692, 9.462781180080377774710511865216, 10.28373772992894348601119755475

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