Properties

Label 2-768-96.59-c1-0-0
Degree $2$
Conductor $768$
Sign $-0.869 + 0.494i$
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  + (−1.44 − 0.956i)3-s + (−1.56 + 3.76i)5-s + (−0.838 + 0.838i)7-s + (1.17 + 2.76i)9-s + (0.249 − 0.601i)11-s + (2.05 − 0.852i)13-s + (5.85 − 3.94i)15-s − 3.23·17-s + (−1.47 − 3.56i)19-s + (2.01 − 0.409i)21-s + (−2.58 + 2.58i)23-s + (−8.21 − 8.21i)25-s + (0.948 − 5.10i)27-s + (−3.52 + 1.45i)29-s − 7.63i·31-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)3-s + (−0.697 + 1.68i)5-s + (−0.316 + 0.316i)7-s + (0.390 + 0.920i)9-s + (0.0750 − 0.181i)11-s + (0.570 − 0.236i)13-s + (1.51 − 1.01i)15-s − 0.785·17-s + (−0.339 − 0.818i)19-s + (0.439 − 0.0893i)21-s + (−0.538 + 0.538i)23-s + (−1.64 − 1.64i)25-s + (0.182 − 0.983i)27-s + (−0.653 + 0.270i)29-s − 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.869 + 0.494i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00129025 - 0.00487964i\)
\(L(\frac12)\) \(\approx\) \(0.00129025 - 0.00487964i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 0.956i)T \)
good5 \( 1 + (1.56 - 3.76i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.838 - 0.838i)T - 7iT^{2} \)
11 \( 1 + (-0.249 + 0.601i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.05 + 0.852i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
19 \( 1 + (1.47 + 3.56i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.58 - 2.58i)T - 23iT^{2} \)
29 \( 1 + (3.52 - 1.45i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.63iT - 31T^{2} \)
37 \( 1 + (0.579 + 0.239i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.54 + 3.54i)T + 41iT^{2} \)
43 \( 1 + (-3.19 - 1.32i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 5.96iT - 47T^{2} \)
53 \( 1 + (-0.762 - 0.315i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.86 + 2.43i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.68 + 6.47i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.78 - 1.98i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (10.2 + 10.2i)T + 71iT^{2} \)
73 \( 1 + (-8.09 + 8.09i)T - 73iT^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (0.998 - 0.413i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-10.4 + 10.4i)T - 89iT^{2} \)
97 \( 1 - 9.21T + 97T^{2} \)
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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

−11.03421844617102479550399639007, −10.33376551985272556160665014695, −9.161383379269871933748188140016, −7.86801715860881594171959046937, −7.29949238660516521607028061029, −6.37476406341598204507016475742, −5.95365325861502316714619230792, −4.43073959891161213287739320467, −3.29482230535386055545634109200, −2.18771868570433971555288116096, 0.00294477344841484607310642282, 1.40100378641047128302870539568, 3.76672368556610859096530855719, 4.32383594786937912668212461240, 5.18151359943823762378459700884, 6.11638795824956332083806476154, 7.15658742581221528741688555368, 8.398006203792683574206343757763, 8.904558804460241230466715033013, 9.850577861784827337415752736944

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