↑

Properties

Label 2-768-32.29-c1-0-4
Degree $2$
Conductor $768$
Sign $0.910 - 0.413i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.705 − 1.70i)5-s + (−3.24 + 3.24i)7-s + (0.707 + 0.707i)9-s + (3.38 − 1.40i)11-s + (0.503 − 1.21i)13-s + 1.84i·15-s + 0.622i·17-s + (−2.14 + 5.17i)19-s + (4.23 − 1.75i)21-s + (2.47 + 2.47i)23-s + (1.13 − 1.13i)25-s + (−0.382 − 0.923i)27-s + (2.16 + 0.897i)29-s + 10.4·31-s + ⋯
L(s)  = 1  + (−0.533 − 0.220i)3-s + (−0.315 − 0.762i)5-s + (−1.22 + 1.22i)7-s + (0.235 + 0.235i)9-s + (1.02 − 0.423i)11-s + (0.139 − 0.337i)13-s + 0.476i·15-s + 0.151i·17-s + (−0.491 + 1.18i)19-s + (0.924 − 0.382i)21-s + (0.516 + 0.516i)23-s + (0.226 − 0.226i)25-s + (−0.0736 − 0.177i)27-s + (0.402 + 0.166i)29-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00011 + 0.216442i\)
\(L(\frac12)\) \(\approx\) \(1.00011 + 0.216442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (0.705 + 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.24 - 3.24i)T - 7iT^{2} \)
11 \( 1 + (-3.38 + 1.40i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.503 + 1.21i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.622iT - 17T^{2} \)
19 \( 1 + (2.14 - 5.17i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.47 - 2.47i)T + 23iT^{2} \)
29 \( 1 + (-2.16 - 0.897i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (-0.0714 - 0.172i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-8.50 - 8.50i)T + 41iT^{2} \)
43 \( 1 + (3.62 - 1.50i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 5.02iT - 47T^{2} \)
53 \( 1 + (-7.15 + 2.96i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.52 + 3.68i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.07 - 1.27i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.17 + 0.901i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.11 + 1.11i)T - 71iT^{2} \)
73 \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + (4.69 - 11.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.54 - 3.54i)T - 89iT^{2} \)
97 \( 1 - 0.139T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.27250080415992660827714723232, −9.468933559819003580416822611462, −8.701647608245542552478691453582, −7.993422642410175926710836174158, −6.44792846825539842408700824560, −6.19165521779763314072467414700, −5.12940063354709992844160983032, −3.95715018521571938973756609294, −2.79146329709640162380715486364, −1.08247139681727881189071599096, 0.71257592426231172653456191445, 2.82079795957737191775936343242, 3.89996650911634404847191480079, 4.56478081779527637277847900584, 6.16580249279447627978945253848, 6.95652747549358562019581423947, 7.08696008318947677279500902809, 8.731204569647475582144415575740, 9.622026033569134209219482944131, 10.34575262916873781226933845657

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