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Properties

Label 2-768-32.29-c1-0-5
Degree $2$
Conductor $768$
Sign $-0.294 - 0.955i$
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 + (1.35 + 3.27i)5-s + (−2.48 + 2.48i)7-s + (0.707 + 0.707i)9-s + (0.420 − 0.174i)11-s + (1.98 − 4.80i)13-s + 3.54i·15-s + 4.75i·17-s + (−0.402 + 0.971i)19-s + (−3.24 + 1.34i)21-s + (−0.739 − 0.739i)23-s + (−5.36 + 5.36i)25-s + (0.382 + 0.923i)27-s + (0.153 + 0.0634i)29-s − 8.57·31-s + ⋯
L(s)  = 1  + (0.533 + 0.220i)3-s + (0.607 + 1.46i)5-s + (−0.939 + 0.939i)7-s + (0.235 + 0.235i)9-s + (0.126 − 0.0525i)11-s + (0.551 − 1.33i)13-s + 0.916i·15-s + 1.15i·17-s + (−0.0923 + 0.222i)19-s + (−0.708 + 0.293i)21-s + (−0.154 − 0.154i)23-s + (−1.07 + 1.07i)25-s + (0.0736 + 0.177i)27-s + (0.0284 + 0.0117i)29-s − 1.54·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.294 - 0.955i$
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: \((2,\ 768,\ (\ :1/2),\ -0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03361 + 1.40034i\)
\(L(\frac12)\) \(\approx\) \(1.03361 + 1.40034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 3.27i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.48 - 2.48i)T - 7iT^{2} \)
11 \( 1 + (-0.420 + 0.174i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.98 + 4.80i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.75iT - 17T^{2} \)
19 \( 1 + (0.402 - 0.971i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.739 + 0.739i)T + 23iT^{2} \)
29 \( 1 + (-0.153 - 0.0634i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 8.57T + 31T^{2} \)
37 \( 1 + (-2.67 - 6.46i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.39 + 1.39i)T + 41iT^{2} \)
43 \( 1 + (-2.84 + 1.18i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.715iT - 47T^{2} \)
53 \( 1 + (-10.4 + 4.32i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.09 - 5.05i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.81 - 1.16i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.39 + 2.23i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-8.26 + 8.26i)T - 71iT^{2} \)
73 \( 1 + (-4.37 - 4.37i)T + 73iT^{2} \)
79 \( 1 - 9.46iT - 79T^{2} \)
83 \( 1 + (-2.85 + 6.88i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.60 + 8.60i)T - 89iT^{2} \)
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

−10.40697334268477044656837802869, −9.887098468379919763404783098025, −8.935760284486333485925830773585, −8.097812484948278087130741202021, −6.99866840601325637304560691930, −6.11187938429117917931301967888, −5.60223989905353072316307936842, −3.69715534940305541990075274999, −3.06461636532767487786272208242, −2.13025288215281529208286332063, 0.833611564521676863550423165573, 2.09063584891283159072476260040, 3.67810937565984476037489816812, 4.47105245564827633562432894919, 5.60319758754243670049129617125, 6.69391124524808010397422755307, 7.40654557821866730607937145728, 8.609495540736828739476709806578, 9.384769604592365041630074655345, 9.555421447537456211677568316046

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