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Properties

Label 2-768-32.29-c1-0-12
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: $\chi_{768} (673, \cdot )$
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

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

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