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Properties

Label 2-24e2-576.205-c1-0-7
Degree $2$
Conductor $576$
Sign $0.0937 - 0.995i$
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  + (0.782 − 1.17i)2-s + (−0.422 + 1.67i)3-s + (−0.775 − 1.84i)4-s + (−0.0860 − 0.0981i)5-s + (1.64 + 1.81i)6-s + (−3.69 − 0.486i)7-s + (−2.77 − 0.528i)8-s + (−2.64 − 1.41i)9-s + (−0.182 + 0.0246i)10-s + (2.51 + 5.09i)11-s + (3.42 − 0.524i)12-s + (2.15 + 6.35i)13-s + (−3.46 + 3.96i)14-s + (0.201 − 0.103i)15-s + (−2.79 + 2.85i)16-s + (−0.490 − 0.490i)17-s + ⋯
L(s)  = 1  + (0.553 − 0.833i)2-s + (−0.243 + 0.969i)3-s + (−0.387 − 0.921i)4-s + (−0.0384 − 0.0438i)5-s + (0.673 + 0.739i)6-s + (−1.39 − 0.183i)7-s + (−0.982 − 0.186i)8-s + (−0.881 − 0.472i)9-s + (−0.0578 + 0.00778i)10-s + (0.757 + 1.53i)11-s + (0.988 − 0.151i)12-s + (0.598 + 1.76i)13-s + (−0.925 + 1.06i)14-s + (0.0519 − 0.0266i)15-s + (−0.699 + 0.714i)16-s + (−0.118 − 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0937 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0937 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0937 - 0.995i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.0937 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680454 + 0.619380i\)
\(L(\frac12)\) \(\approx\) \(0.680454 + 0.619380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.782 + 1.17i)T \)
3 \( 1 + (0.422 - 1.67i)T \)
good5 \( 1 + (0.0860 + 0.0981i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (3.69 + 0.486i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (-2.51 - 5.09i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-2.15 - 6.35i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (0.490 + 0.490i)T + 17iT^{2} \)
19 \( 1 + (0.255 - 1.28i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.753 - 5.72i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (7.24 - 0.474i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (4.69 - 2.70i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.531 - 2.67i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.02 + 7.81i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (2.75 - 1.35i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (3.95 - 1.06i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.43 + 5.14i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (3.69 + 4.21i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.118 - 1.81i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-7.81 - 3.85i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (2.55 + 6.17i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.06 + 7.40i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.42 - 16.4i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.87 - 2.52i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (-6.87 + 2.84i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-6.08 - 3.51i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97297951790071448764887063477, −9.953466637056559698101938357403, −9.471253784019677659930072522895, −8.997045133097427006847615244922, −6.90719586165565953458792231059, −6.29605066349899439213122501443, −5.06880837037298916754008328966, −4.01117166926622533601673396485, −3.59339079041325144608297309679, −1.93376568900276421272098851599, 0.43584080151005294282299060656, 2.96796725628580911666312672244, 3.57022211446332027803880277600, 5.48167872791592029799422592184, 6.03050845608822806626485178795, 6.65518252647380353374923226601, 7.68508273559815719773014878680, 8.541062846644946876384943550926, 9.250264015904511029336799110965, 10.77688437333653644657954955327

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