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Properties

Label 2-768-96.35-c1-0-23
Degree $2$
Conductor $768$
Sign $-0.0106 + 0.999i$
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  + (0.266 − 1.71i)3-s + (3.14 − 1.30i)5-s + (0.663 + 0.663i)7-s + (−2.85 − 0.911i)9-s + (−1.91 + 0.794i)11-s + (2.31 − 5.59i)13-s + (−1.39 − 5.73i)15-s + 2.24·17-s + (−3.08 − 1.27i)19-s + (1.31 − 0.958i)21-s + (4.32 + 4.32i)23-s + (4.66 − 4.66i)25-s + (−2.32 + 4.64i)27-s + (−0.546 + 1.32i)29-s − 2.34i·31-s + ⋯
L(s)  = 1  + (0.153 − 0.988i)3-s + (1.40 − 0.582i)5-s + (0.250 + 0.250i)7-s + (−0.952 − 0.303i)9-s + (−0.578 + 0.239i)11-s + (0.642 − 1.55i)13-s + (−0.359 − 1.48i)15-s + 0.545·17-s + (−0.708 − 0.293i)19-s + (0.286 − 0.209i)21-s + (0.901 + 0.901i)23-s + (0.933 − 0.933i)25-s + (−0.446 + 0.894i)27-s + (−0.101 + 0.245i)29-s − 0.420i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.0106 + 0.999i$
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: \((2,\ 768,\ (\ :1/2),\ -0.0106 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40282 - 1.41790i\)
\(L(\frac12)\) \(\approx\) \(1.40282 - 1.41790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.266 + 1.71i)T \)
good5 \( 1 + (-3.14 + 1.30i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.663 - 0.663i)T + 7iT^{2} \)
11 \( 1 + (1.91 - 0.794i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.31 + 5.59i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + (3.08 + 1.27i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.32 - 4.32i)T + 23iT^{2} \)
29 \( 1 + (0.546 - 1.32i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.34iT - 31T^{2} \)
37 \( 1 + (-0.324 - 0.783i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.73 - 4.73i)T - 41iT^{2} \)
43 \( 1 + (0.951 + 2.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 + (-3.49 - 8.43i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.11 + 7.51i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.01 - 0.421i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.46 - 8.35i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-0.167 + 0.167i)T - 71iT^{2} \)
73 \( 1 + (-3.86 - 3.86i)T + 73iT^{2} \)
79 \( 1 - 2.44T + 79T^{2} \)
83 \( 1 + (-5.17 + 12.4i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.63 - 6.63i)T + 89iT^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−10.08190619581776800500306053965, −9.120296077320910838575580874249, −8.391680502117312011613603736327, −7.62625949711936094111557921457, −6.48844062111522355941255971380, −5.59659029813088437837207694426, −5.18265106353799606637955814429, −3.19740478338094636282991996181, −2.14954449128084023709266949496, −1.06403742528734171507783377845, 1.88881829749499259706275568706, 2.96395535529757688908243764580, 4.17207591341554077584863715076, 5.16679621814011836063496731753, 6.08196230317963375235886922423, 6.82240156983106048304733581932, 8.225534360787588924706476345600, 9.070860610190725604876768966582, 9.689024442671811729478448288847, 10.72341417949423802168492973297

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