↑

Properties

Label 2-24e2-144.11-c1-0-0
Degree $2$
Conductor $576$
Sign $-0.957 + 0.289i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (−1.67 + 0.427i)3-s + (−0.170 − 0.0458i)5-s + (−1.17 + 2.03i)7-s + (2.63 − 1.43i)9-s + (0.340 − 0.0913i)11-s + (−1.49 − 0.399i)13-s + (0.306 + 0.00382i)15-s + 3.58i·17-s + (−5.36 − 5.36i)19-s + (1.10 − 3.91i)21-s + (−0.165 + 0.0953i)23-s + (−4.30 − 2.48i)25-s + (−3.80 + 3.53i)27-s + (−9.10 + 2.43i)29-s + (3.43 − 1.98i)31-s + ⋯
L(s)  = 1  + (−0.969 + 0.246i)3-s + (−0.0764 − 0.0204i)5-s + (−0.443 + 0.768i)7-s + (0.878 − 0.478i)9-s + (0.102 − 0.0275i)11-s + (−0.413 − 0.110i)13-s + (0.0791 + 0.000988i)15-s + 0.868i·17-s + (−1.23 − 1.23i)19-s + (0.240 − 0.854i)21-s + (−0.0344 + 0.0198i)23-s + (−0.860 − 0.496i)25-s + (−0.733 + 0.680i)27-s + (−1.69 + 0.452i)29-s + (0.617 − 0.356i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00514617 - 0.0347705i\)
\(L(\frac12)\) \(\approx\) \(0.00514617 - 0.0347705i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.427i)T \)
good5 \( 1 + (0.170 + 0.0458i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.17 - 2.03i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.340 + 0.0913i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.49 + 0.399i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.58iT - 17T^{2} \)
19 \( 1 + (5.36 + 5.36i)T + 19iT^{2} \)
23 \( 1 + (0.165 - 0.0953i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (9.10 - 2.43i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.43 + 1.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.28 + 3.28i)T + 37iT^{2} \)
41 \( 1 + (4.25 + 7.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.09 + 4.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.93 - 8.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.83 - 4.83i)T - 53iT^{2} \)
59 \( 1 + (-0.720 + 2.68i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.13 - 7.97i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.06 + 11.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.13iT - 71T^{2} \)
73 \( 1 + 5.67iT - 73T^{2} \)
79 \( 1 + (12.8 + 7.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.31 - 12.3i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 + (0.996 - 1.72i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.12953504048554243950604138204, −10.47837394365503767134236806848, −9.489114786100797673656029409625, −8.774216850384687362841715723146, −7.50271497712333664779717366636, −6.45618196052650162596348319168, −5.80789160831648895805374262928, −4.77763440461568393924839209457, −3.71970733557632465563048848715, −2.11234507385087348569969434143, 0.02141381067112564801408144076, 1.77973401801643323876632242085, 3.62346335273347516323642027701, 4.62643636448473803739156846412, 5.71186699717976232415369660097, 6.64649260477970434476751713769, 7.34100473485781461401099801116, 8.308895028645748700142116261329, 9.854455438033856745660018812242, 10.08167771719577482452402689688

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