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Properties

Label 1-1476-1476.1267-r1-0-0
Degree $1$
Conductor $1476$
Sign $0.952 - 0.306i$
Analytic cond. $158.618$
Root an. cond. $158.618$
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  + (0.913 − 0.406i)5-s + (−0.669 + 0.743i)7-s + (−0.913 − 0.406i)11-s + (0.669 + 0.743i)13-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.669 − 0.743i)25-s + (−0.104 + 0.994i)29-s + (0.104 + 0.994i)31-s + (−0.309 + 0.951i)35-s + (−0.809 − 0.587i)37-s + (0.978 − 0.207i)43-s + (0.978 − 0.207i)47-s + (−0.104 − 0.994i)49-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)5-s + (−0.669 + 0.743i)7-s + (−0.913 − 0.406i)11-s + (0.669 + 0.743i)13-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.669 − 0.743i)25-s + (−0.104 + 0.994i)29-s + (0.104 + 0.994i)31-s + (−0.309 + 0.951i)35-s + (−0.809 − 0.587i)37-s + (0.978 − 0.207i)43-s + (0.978 − 0.207i)47-s + (−0.104 − 0.994i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $0.952 - 0.306i$
Analytic conductor: \(158.618\)
Root analytic conductor: \(158.618\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1476,\ (1:\ ),\ 0.952 - 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.843724263 - 0.2890421051i\)
\(L(\frac12)\) \(\approx\) \(1.843724263 - 0.2890421051i\)
\(L(1)\) \(\approx\) \(1.070647844 + 0.01310561698i\)
\(L(1)\) \(\approx\) \(1.070647844 + 0.01310561698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.978 - 0.207i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.73597331680770210640170661966, −19.866443105619792694830758683467, −18.86161892893052260097700966828, −18.27034676237022902128254260356, −17.49952497074675908544639830843, −16.942030829188479820294962082935, −15.83317307029810216793049347999, −15.434622594126432589151159555146, −14.30174111429480014614311636113, −13.44448289427592097115715570668, −13.272854142578078973456366217347, −12.26184676565390799604978129386, −11.08241331874160519444929060516, −10.38996759791060343580707758290, −9.911678369861235109665943865293, −9.087294593626356668533010333842, −7.88096210251586417721304686228, −7.29930688234857006865993101799, −6.171500116947251433913057221243, −5.81641516565984928557262953094, −4.6171924984625084269527510386, −3.6381767312336390364610392386, −2.73136449034374639709972960460, −1.88253420947117499269159638237, −0.635960341364984942372627192960, 0.506490892967892132093447325002, 1.88748936446424016946945141881, 2.4778157003116634811824534216, 3.54803098471291870650159369388, 4.7225364295853869299884995819, 5.48301475805974565366774945528, 6.30996824959664534160341479390, 6.83719934670332312885873714814, 8.34193941477869400658887942067, 8.84823373139041524735096357283, 9.4658609831967797969849564008, 10.555566025891407812867685909524, 11.0152746034929505130028785561, 12.393996942955528266486599981153, 12.73963806315320078427738197131, 13.622999377318644313549942674393, 14.16006213279609270242430892082, 15.36578110147419858182629343272, 15.96876575012186055955906189049, 16.54941072034223787031907391655, 17.59056448271289042588735731053, 18.1440770585465946932941860890, 18.88754764436799433199573150200, 19.64959053886118237526947537734, 20.56790755269030075152163379905

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