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Properties

Label 2-1472-1472.1333-c0-0-2
Degree $2$
Conductor $1472$
Sign $0.881 - 0.471i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.608 + 0.793i)2-s + (1.85 − 0.369i)3-s + (−0.258 − 0.965i)4-s + (−0.837 + 1.69i)6-s + (0.923 + 0.382i)8-s + (2.38 − 0.989i)9-s + (−0.837 − 1.69i)12-s + (−0.835 + 1.25i)13-s + (−0.866 + 0.499i)16-s + (−0.669 + 2.49i)18-s + (0.382 + 0.923i)23-s + (1.85 + 0.369i)24-s + (−0.382 + 0.923i)25-s + (−0.483 − 1.42i)26-s + (2.49 − 1.66i)27-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (1.85 − 0.369i)3-s + (−0.258 − 0.965i)4-s + (−0.837 + 1.69i)6-s + (0.923 + 0.382i)8-s + (2.38 − 0.989i)9-s + (−0.837 − 1.69i)12-s + (−0.835 + 1.25i)13-s + (−0.866 + 0.499i)16-s + (−0.669 + 2.49i)18-s + (0.382 + 0.923i)23-s + (1.85 + 0.369i)24-s + (−0.382 + 0.923i)25-s + (−0.483 − 1.42i)26-s + (2.49 − 1.66i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470643910\)
\(L(\frac12)\) \(\approx\) \(1.470643910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 - 0.793i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
good3 \( 1 + (-1.85 + 0.369i)T + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (0.835 - 1.25i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.349 + 1.75i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + 1.21iT - T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
53 \( 1 + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.241 + 0.0999i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (1.78 - 0.739i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.546773395626280512113391449201, −8.973522351024259502556943889813, −8.069587177754238668904114290984, −7.48744838397511229860707234320, −7.00511632044461169667171000646, −5.94693816373854407261830375242, −4.59483017604875724778030544075, −3.78915855650649696397254590741, −2.41548845763799947660124788246, −1.63597067266958237522364988625, 1.58692141040650096984411319709, 2.78908988993675154597253490537, 3.10384323064120358332690988212, 4.21329855143058927615634598832, 5.03694543484856123926247836760, 6.95209497977235106579876180787, 7.58780029028992043375632052718, 8.387806354186008501290613260238, 8.780388821236255327996603249342, 9.581395611964288569097073096171

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