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Properties

Label 2-1287-1287.571-c0-0-7
Degree $2$
Conductor $1287$
Sign $0.990 + 0.139i$
Analytic cond. $0.642296$
Root an. cond. $0.801434$
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  + (0.104 + 0.181i)2-s + (0.913 + 0.406i)3-s + (0.478 − 0.828i)4-s + (0.0218 + 0.207i)6-s + (−0.309 − 0.535i)7-s + 0.408·8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)11-s + (0.773 − 0.562i)12-s + (−0.5 + 0.866i)13-s + (0.0646 − 0.111i)14-s + (−0.435 − 0.754i)16-s + (−0.0646 + 0.198i)18-s + 1.82·19-s + (−0.0646 − 0.614i)21-s + (0.104 − 0.181i)22-s + ⋯
L(s)  = 1  + (0.104 + 0.181i)2-s + (0.913 + 0.406i)3-s + (0.478 − 0.828i)4-s + (0.0218 + 0.207i)6-s + (−0.309 − 0.535i)7-s + 0.408·8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)11-s + (0.773 − 0.562i)12-s + (−0.5 + 0.866i)13-s + (0.0646 − 0.111i)14-s + (−0.435 − 0.754i)16-s + (−0.0646 + 0.198i)18-s + 1.82·19-s + (−0.0646 − 0.614i)21-s + (0.104 − 0.181i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.629778312\)
\(L(\frac12)\) \(\approx\) \(1.629778312\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.104 - 0.181i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.82T + T^{2} \)
23 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.770574518438888885174533885389, −9.321104073025394397405442354868, −8.061914094276498997692970277783, −7.48482847517312243074503625472, −6.62418744683921802222755722612, −5.59335500750087981927581910925, −4.77779835307372640127232887604, −3.67358566140760172718686615970, −2.73680576844090356732339819180, −1.50815942081390644685327369165, 1.86917076283186324895825586996, 2.79776062286001112699270582209, 3.38941890690280704504717466305, 4.59511453957219050673753250283, 5.75829877030877188212738421033, 6.94091342310603214849330489105, 7.55634115337769427092908863973, 8.048016663253287162100377707480, 9.027638766991866294615565443229, 9.786312890542851399094209901251

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