Properties

Label 2-1287-143.51-c0-0-1
Degree $2$
Conductor $1287$
Sign $0.390 + 0.920i$
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  + (−1.59 − 1.16i)2-s + (0.896 + 2.76i)4-s + (0.533 + 0.734i)5-s + (1.16 − 3.57i)8-s − 1.79i·10-s + (0.156 − 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (−1.54 + 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (−1.87 + 0.610i)26-s + 5.17·32-s + (3.24 − 1.05i)40-s + (−0.437 + 1.34i)41-s + ⋯
L(s)  = 1  + (−1.59 − 1.16i)2-s + (0.896 + 2.76i)4-s + (0.533 + 0.734i)5-s + (1.16 − 3.57i)8-s − 1.79i·10-s + (0.156 − 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (−1.54 + 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (−1.87 + 0.610i)26-s + 5.17·32-s + (3.24 − 1.05i)40-s + (−0.437 + 1.34i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ 0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5703491641\)
\(L(\frac12)\) \(\approx\) \(0.5703491641\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
good2 \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.90iT - T^{2} \)
47 \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.893081671995599104092371788028, −8.914045680330983540658754696614, −8.457174160912861038078164361818, −7.56353807162850211583288747174, −6.73313180351450643162329403311, −5.80933776318411123060605110988, −3.95205261976299428233217373905, −3.10417873952285042535252860305, −2.35073451998344739498823561280, −0.999076212008781096268224931793, 1.26012727978784758800376288542, 2.13701726387964526188768106087, 4.42970152119008982492506979317, 5.34721213924123730317942115531, 6.10952130737628475332599994960, 6.94444698675918518831893984650, 7.57425033446967410796846623841, 8.593341214875123791247105271074, 9.075189892482585387047379942537, 9.652703120877793628929443326828

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