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Properties

Label 2-768-96.35-c1-0-22
Degree $2$
Conductor $768$
Sign $-0.468 + 0.883i$
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  + (1.02 − 1.39i)3-s + (−3.14 + 1.30i)5-s + (0.663 + 0.663i)7-s + (−0.911 − 2.85i)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 + (1.60 − 0.249i)21-s + (−4.32 − 4.32i)23-s + (4.66 − 4.66i)25-s + (−4.92 − 1.64i)27-s + (0.546 − 1.32i)29-s − 2.34i·31-s + ⋯
L(s)  = 1  + (0.589 − 0.807i)3-s + (−1.40 + 0.582i)5-s + (0.250 + 0.250i)7-s + (−0.303 − 0.952i)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 + (0.350 − 0.0545i)21-s + (−0.901 − 0.901i)23-s + (0.933 − 0.933i)25-s + (−0.948 − 0.316i)27-s + (0.101 − 0.245i)29-s − 0.420i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.468 + 0.883i$
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: \((2,\ 768,\ (\ :1/2),\ -0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589925 - 0.980337i\)
\(L(\frac12)\) \(\approx\) \(0.589925 - 0.980337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 1.39i)T \)
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

−10.14824895757219364425889613808, −8.688723409484796387360622793048, −8.339419550264856620473485220491, −7.57631099400618971230595144403, −6.71952103175314603286563553450, −5.86133895387846739552255234538, −4.21449494088305022713260995549, −3.43393714788070295336520960624, −2.38557556301656686748550721840, −0.54591454654949817040274835112, 1.75811697374312950474929826305, 3.51857769868793707600688260092, 4.23504644505828270394367646411, 4.66741483658773128113018483363, 6.25907014037780921978530037233, 7.42218997149163790829763200265, 8.173036919888027754955809232431, 8.905177422486539885795453367240, 9.493589863894976797692369848114, 10.72066618119213331614123973009

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