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Properties

Label 2-287-287.286-c2-0-24
Degree $2$
Conductor $287$
Sign $0.801 - 0.597i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4.37·3-s − 3·4-s − 3.17i·5-s − 4.37·6-s + (−0.818 + 6.95i)7-s + 7·8-s + 10.1·9-s + 3.17i·10-s + 19.4i·11-s − 13.1·12-s − 4.37·13-s + (0.818 − 6.95i)14-s − 13.9i·15-s + 5·16-s + 25.3·17-s + ⋯
L(s)  = 1  − 0.5·2-s + 1.45·3-s − 0.750·4-s − 0.635i·5-s − 0.729·6-s + (−0.116 + 0.993i)7-s + 0.875·8-s + 1.12·9-s + 0.317i·10-s + 1.76i·11-s − 1.09·12-s − 0.336·13-s + (0.0584 − 0.496i)14-s − 0.926i·15-s + 0.312·16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.62644 + 0.539830i\)
\(L(\frac12)\) \(\approx\) \(1.62644 + 0.539830i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.818 - 6.95i)T \)
41 \( 1 + (-28.1 - 29.7i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 - 4.37T + 9T^{2} \)
5 \( 1 + 3.17iT - 25T^{2} \)
11 \( 1 - 19.4iT - 121T^{2} \)
13 \( 1 + 4.37T + 169T^{2} \)
17 \( 1 - 25.3T + 289T^{2} \)
19 \( 1 - 23.3T + 361T^{2} \)
23 \( 1 - 0.747T + 529T^{2} \)
29 \( 1 + 11.3iT - 841T^{2} \)
31 \( 1 + 17.7iT - 961T^{2} \)
37 \( 1 + 37.1T + 1.36e3T^{2} \)
43 \( 1 - 35.4T + 1.84e3T^{2} \)
47 \( 1 + 52.0T + 2.20e3T^{2} \)
53 \( 1 - 69.2iT - 2.80e3T^{2} \)
59 \( 1 - 20.9iT - 3.48e3T^{2} \)
61 \( 1 - 75.4iT - 3.72e3T^{2} \)
67 \( 1 + 61.1iT - 4.48e3T^{2} \)
71 \( 1 + 107. iT - 5.04e3T^{2} \)
73 \( 1 - 19.0iT - 5.32e3T^{2} \)
79 \( 1 - 46.9iT - 6.24e3T^{2} \)
83 \( 1 + 165. iT - 6.88e3T^{2} \)
89 \( 1 + 70.9T + 7.92e3T^{2} \)
97 \( 1 - 100.T + 9.40e3T^{2} \)
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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

−12.07855682239202690720881174819, −10.08627836156027687551490863214, −9.474720840246410246181368599211, −9.045420515580751340221950576199, −7.969308957173813728076218314628, −7.45450464740132194865138167698, −5.39108232544212100472307221215, −4.41595303150734273438275784563, −2.99240006365398813832655668845, −1.58259101447336891926685299606, 1.00842308874681905703566984295, 3.18679382191528149114816429107, 3.62931433385231657935234952004, 5.34184589417762065170149194016, 7.08223139297054363711633325270, 7.892729954000618277170305767542, 8.565963773852478659080298043686, 9.543643276033569719341771829534, 10.23495288367159389077867842548, 11.16333981883852669337574056046

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