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Properties

Label 2-24e2-16.13-c1-0-7
Degree $2$
Conductor $576$
Sign $0.382 + 0.923i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)5-s − 2i·7-s + (1 − i)11-s + (−1 − i)13-s + 2·17-s + (−3 − 3i)19-s − 6i·23-s + 3i·25-s + (−3 − 3i)29-s + 8·31-s + (−2 − 2i)35-s + (3 − 3i)37-s + (−5 + 5i)43-s + 8·47-s + 3·49-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s − 0.755i·7-s + (0.301 − 0.301i)11-s + (−0.277 − 0.277i)13-s + 0.485·17-s + (−0.688 − 0.688i)19-s − 1.25i·23-s + 0.600i·25-s + (−0.557 − 0.557i)29-s + 1.43·31-s + (−0.338 − 0.338i)35-s + (0.493 − 0.493i)37-s + (−0.762 + 0.762i)43-s + 1.16·47-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24354 - 0.830910i\)
\(L(\frac12)\) \(\approx\) \(1.24354 - 0.830910i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (3 - 3i)T - 59iT^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.49457362522048707912170185274, −9.738002026339452232707527063661, −8.827648173254806009050575758447, −7.952217381380142089495914242612, −6.92280067119968710436887352894, −6.01316137671141209065033266010, −4.90345101020265287924670550326, −3.96234967114717235772572934758, −2.53296449977388423632414205611, −0.896756672785785301727100718539, 1.79537193214410255519390090922, 2.94986240719238815556388368069, 4.27159379593098255531242198801, 5.52372752609787966568192207742, 6.26462668120333481933310169658, 7.26456772043254885247206773098, 8.291743091132158793423402148583, 9.238224148555721332799289097895, 9.985492334308252459548449454653, 10.77311636803085939777919533946

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