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Properties

Label 2-768-32.13-c1-0-13
Degree $2$
Conductor $768$
Sign $-0.956 + 0.291i$
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  + (0.382 − 0.923i)3-s + (−1.60 + 0.666i)5-s + (−0.589 + 0.589i)7-s + (−0.707 − 0.707i)9-s + (0.657 + 1.58i)11-s + (−3.87 − 1.60i)13-s + 1.74i·15-s − 7.96i·17-s + (−3.97 − 1.64i)19-s + (0.319 + 0.770i)21-s + (−0.452 − 0.452i)23-s + (−1.39 + 1.39i)25-s + (−0.923 + 0.382i)27-s + (1.69 − 4.08i)29-s − 9.32·31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (−0.719 + 0.298i)5-s + (−0.222 + 0.222i)7-s + (−0.235 − 0.235i)9-s + (0.198 + 0.478i)11-s + (−1.07 − 0.444i)13-s + 0.449i·15-s − 1.93i·17-s + (−0.911 − 0.377i)19-s + (0.0696 + 0.168i)21-s + (−0.0942 − 0.0942i)23-s + (−0.278 + 0.278i)25-s + (−0.177 + 0.0736i)27-s + (0.314 − 0.758i)29-s − 1.67·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.956 + 0.291i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.956 + 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0625512 - 0.419621i\)
\(L(\frac12)\) \(\approx\) \(0.0625512 - 0.419621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (1.60 - 0.666i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.589 - 0.589i)T - 7iT^{2} \)
11 \( 1 + (-0.657 - 1.58i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.87 + 1.60i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 7.96iT - 17T^{2} \)
19 \( 1 + (3.97 + 1.64i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.452 + 0.452i)T + 23iT^{2} \)
29 \( 1 + (-1.69 + 4.08i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 9.32T + 31T^{2} \)
37 \( 1 + (0.810 - 0.335i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.65 + 6.65i)T + 41iT^{2} \)
43 \( 1 + (-2.22 - 5.36i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.50iT - 47T^{2} \)
53 \( 1 + (1.10 + 2.67i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.92 + 0.796i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.70 + 11.3i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (4.54 - 10.9i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-9.09 + 9.09i)T - 71iT^{2} \)
73 \( 1 + (-1.65 - 1.65i)T + 73iT^{2} \)
79 \( 1 - 0.580iT - 79T^{2} \)
83 \( 1 + (3.33 + 1.38i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.91 - 4.91i)T - 89iT^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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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

−9.739506242553925242302829294386, −9.180991392095709427897118797416, −8.004126828157410888211412788681, −7.32592340478483241262499374438, −6.74592764204433698975706273486, −5.43637909052224766727761432703, −4.43334497172097433532193428427, −3.16172322148591645817448462474, −2.22527071162834998097203469778, −0.19504814947577841525029198685, 1.97009904788294318379362357182, 3.59750616558135061040815457664, 4.10203135503739671031479931919, 5.23471706420948213405388089425, 6.33587794998313016004689349122, 7.35186993023292945469551966018, 8.367480906111460609234395653093, 8.811443535711605968187932887756, 10.00067858804814125110361096175, 10.55077543669776616758012823301

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