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Properties

Label 2-768-96.35-c1-0-12
Degree $2$
Conductor $768$
Sign $0.704 - 0.709i$
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  + (1.29 + 1.15i)3-s + (−0.180 + 0.0746i)5-s + (−0.289 − 0.289i)7-s + (0.338 + 2.98i)9-s + (3.10 − 1.28i)11-s + (1.27 − 3.08i)13-s + (−0.318 − 0.111i)15-s + 0.806·17-s + (5.57 + 2.31i)19-s + (−0.0400 − 0.706i)21-s + (5.03 + 5.03i)23-s + (−3.50 + 3.50i)25-s + (−3.00 + 4.24i)27-s + (−3.64 + 8.78i)29-s − 7.48i·31-s + ⋯
L(s)  = 1  + (0.745 + 0.666i)3-s + (−0.0805 + 0.0333i)5-s + (−0.109 − 0.109i)7-s + (0.112 + 0.993i)9-s + (0.937 − 0.388i)11-s + (0.354 − 0.856i)13-s + (−0.0823 − 0.0287i)15-s + 0.195·17-s + (1.27 + 0.530i)19-s + (−0.00873 − 0.154i)21-s + (1.05 + 1.05i)23-s + (−0.701 + 0.701i)25-s + (−0.577 + 0.816i)27-s + (−0.675 + 1.63i)29-s − 1.34i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91434 + 0.796562i\)
\(L(\frac12)\) \(\approx\) \(1.91434 + 0.796562i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.15i)T \)
good5 \( 1 + (0.180 - 0.0746i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.289 + 0.289i)T + 7iT^{2} \)
11 \( 1 + (-3.10 + 1.28i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.27 + 3.08i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.806T + 17T^{2} \)
19 \( 1 + (-5.57 - 2.31i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.03 - 5.03i)T + 23iT^{2} \)
29 \( 1 + (3.64 - 8.78i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.48iT - 31T^{2} \)
37 \( 1 + (-1.47 - 3.55i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.62 - 2.62i)T - 41iT^{2} \)
43 \( 1 + (2.84 + 6.87i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.399iT - 47T^{2} \)
53 \( 1 + (-1.42 - 3.43i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.918 - 2.21i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (8.81 + 3.64i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.76 + 9.09i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.86 - 2.86i)T - 71iT^{2} \)
73 \( 1 + (6.97 + 6.97i)T + 73iT^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 + (-2.47 + 5.96i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.43 + 5.43i)T + 89iT^{2} \)
97 \( 1 - 2.61T + 97T^{2} \)
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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

−10.29437435983476665912846623250, −9.464791162901975724364384392050, −8.925268320398076915584357050545, −7.86915564540127596142438432809, −7.23292656443888211967015101306, −5.82200597506442987229000343896, −5.03520384915198267705568865065, −3.57201966995379179455581443196, −3.31480656304301549431190276717, −1.50092191587074891542687923294, 1.18453775707299610859122523443, 2.45390805762290461726789538495, 3.60922731518513963715452588394, 4.59554535346863132909102774239, 6.07782139417009230377362524437, 6.83493944922616300136018726284, 7.56704977198085360701349992559, 8.602671973889525201568920872849, 9.243367804714492225347494884897, 9.915339057250294419121885133559

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