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Properties

Label 2-1472-1472.45-c0-0-2
Degree $2$
Conductor $1472$
Sign $0.956 - 0.290i$
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  + (0.793 − 0.608i)2-s + (0.996 + 1.49i)3-s + (0.258 − 0.965i)4-s + (1.69 + 0.576i)6-s + (−0.382 − 0.923i)8-s + (−0.848 + 2.04i)9-s + (1.69 − 0.576i)12-s + (−0.0255 + 0.128i)13-s + (−0.866 − 0.499i)16-s + (0.573 + 2.14i)18-s + (0.923 + 0.382i)23-s + (0.996 − 1.49i)24-s + (−0.923 + 0.382i)25-s + (0.0578 + 0.117i)26-s + (−2.14 + 0.426i)27-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.996 + 1.49i)3-s + (0.258 − 0.965i)4-s + (1.69 + 0.576i)6-s + (−0.382 − 0.923i)8-s + (−0.848 + 2.04i)9-s + (1.69 − 0.576i)12-s + (−0.0255 + 0.128i)13-s + (−0.866 − 0.499i)16-s + (0.573 + 2.14i)18-s + (0.923 + 0.382i)23-s + (0.996 − 1.49i)24-s + (−0.923 + 0.382i)25-s + (0.0578 + 0.117i)26-s + (−2.14 + 0.426i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.212219074\)
\(L(\frac12)\) \(\approx\) \(2.212219074\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-0.996 - 1.49i)T + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (0.0255 - 0.128i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.534 + 0.357i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + 1.58iT - T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.923 + 0.382i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.382 + 0.923i)T^{2} \)
47 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
53 \( 1 + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.739 - 1.78i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.923 - 0.382i)T^{2} \)
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

−9.729888826586958383556505285740, −9.321923383710288727650359690149, −8.400564493487693475555958141878, −7.42269615365605785112358305589, −6.13337314763184219359456288258, −5.18944794286851368948634978607, −4.49184378923829077023423206270, −3.71207868526650239601602077435, −3.00847442449069907432304846725, −2.00174920391195998122248364421, 1.61210529516156052320820393839, 2.76576470196142004438640951467, 3.38715310997106749300321994899, 4.66126978984204951093912704311, 5.76962261611922621203912069702, 6.69260700264179948398905855023, 7.06936667805800211352064037648, 7.976924962315707449962961760406, 8.486196515211007956700834465562, 9.182391304213612287744350284788

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