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Properties

Label 2-768-96.59-c1-0-19
Degree $2$
Conductor $768$
Sign $0.784 + 0.620i$
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  + (−0.345 + 1.69i)3-s + (1.56 − 3.76i)5-s + (−0.838 + 0.838i)7-s + (−2.76 − 1.17i)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 + (−1.13 − 1.71i)21-s + (2.58 − 2.58i)23-s + (−8.21 − 8.21i)25-s + (2.94 − 4.28i)27-s + (3.52 − 1.45i)29-s − 7.63i·31-s + ⋯
L(s)  = 1  + (−0.199 + 0.979i)3-s + (0.697 − 1.68i)5-s + (−0.316 + 0.316i)7-s + (−0.920 − 0.390i)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 + (−0.247 − 0.373i)21-s + (0.538 − 0.538i)23-s + (−1.64 − 1.64i)25-s + (0.566 − 0.824i)27-s + (0.653 − 0.270i)29-s − 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.784 + 0.620i$
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: \((2,\ 768,\ (\ :1/2),\ 0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38674 - 0.482458i\)
\(L(\frac12)\) \(\approx\) \(1.38674 - 0.482458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.345 - 1.69i)T \)
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

−9.972118272036518226286257619667, −9.407851954056764713134540769487, −8.769054243928618417550818882472, −8.041978936304742602442511914021, −6.33583814411108940244392418226, −5.59354723108536142861132952500, −4.86808016882380848303040441367, −4.04128834678086383397849298279, −2.57480442154327836764383382319, −0.822638898671271756752916317389, 1.51108772866595608288821781739, 2.76294149000784066407386923989, 3.55794799407315539129867898592, 5.48005458549230944586858360000, 6.21441224421258638467116575144, 6.88237111692107816585352753573, 7.52176637210764190065611850323, 8.580516579878090864106501141207, 9.769914821380966488878373023087, 10.59282926872459662133296106764

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