Properties

Label 2-1287-143.142-c0-0-0
Degree $2$
Conductor $1287$
Sign $-0.707 - 0.707i$
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  − 1.41·2-s + 1.00·4-s + 1.41i·5-s − 2.00i·10-s + (−0.707 + 0.707i)11-s + i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 − 1.00i)22-s − 1.00·25-s − 1.41i·26-s + 1.41·32-s − 1.41·41-s + (−0.707 + 0.707i)44-s − 1.41i·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s + 1.41i·5-s − 2.00i·10-s + (−0.707 + 0.707i)11-s + i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 − 1.00i)22-s − 1.00·25-s − 1.41i·26-s + 1.41·32-s − 1.41·41-s + (−0.707 + 0.707i)44-s − 1.41i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3784684085\)
\(L(\frac12)\) \(\approx\) \(0.3784684085\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 - iT \)
good2 \( 1 + 1.41T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−10.04390623027777445219893509241, −9.548605721503100219266938815734, −8.532635996048212080025464920340, −7.80517265170894895481697674436, −6.86553503070627652387055000760, −6.73843198351514721437403768470, −5.21323718311498201761764127839, −3.99070595456108553418391363479, −2.68975853155993740255287343170, −1.81985626672467934418022898582, 0.51655883151640336112889117237, 1.67270824432691264593835114969, 3.14015175721554756128060612534, 4.62046663623492253077066128744, 5.32155161019000700325336074465, 6.36065687336432607164610453117, 7.70259549276075207132621123663, 8.055056054838760216502210306324, 8.777036335306468919375663038887, 9.349590651447221870559222340927

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