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Properties

Label 2-287-287.286-c2-0-8
Degree $2$
Conductor $287$
Sign $-0.837 - 0.547i$
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  + 0.517·2-s + 2.41·3-s − 3.73·4-s + 2.97i·5-s + 1.24·6-s + (−2.38 + 6.58i)7-s − 4.00·8-s − 3.16·9-s + 1.54i·10-s − 4.27i·11-s − 9.01·12-s − 14.4·13-s + (−1.23 + 3.40i)14-s + 7.19i·15-s + 12.8·16-s − 14.2·17-s + ⋯
L(s)  = 1  + 0.258·2-s + 0.804·3-s − 0.933·4-s + 0.595i·5-s + 0.208·6-s + (−0.340 + 0.940i)7-s − 0.500·8-s − 0.352·9-s + 0.154i·10-s − 0.388i·11-s − 0.751·12-s − 1.11·13-s + (−0.0880 + 0.243i)14-s + 0.479i·15-s + 0.803·16-s − 0.839·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.837 - 0.547i$
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: \((2,\ 287,\ (\ :1),\ -0.837 - 0.547i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.237233 + 0.796645i\)
\(L(\frac12)\) \(\approx\) \(0.237233 + 0.796645i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.38 - 6.58i)T \)
41 \( 1 + (-9.42 + 39.9i)T \)
good2 \( 1 - 0.517T + 4T^{2} \)
3 \( 1 - 2.41T + 9T^{2} \)
5 \( 1 - 2.97iT - 25T^{2} \)
11 \( 1 + 4.27iT - 121T^{2} \)
13 \( 1 + 14.4T + 169T^{2} \)
17 \( 1 + 14.2T + 289T^{2} \)
19 \( 1 + 7.77T + 361T^{2} \)
23 \( 1 + 5.72T + 529T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 - 47.8iT - 961T^{2} \)
37 \( 1 - 46.7T + 1.36e3T^{2} \)
43 \( 1 - 7.24T + 1.84e3T^{2} \)
47 \( 1 - 1.84T + 2.20e3T^{2} \)
53 \( 1 + 2.40iT - 2.80e3T^{2} \)
59 \( 1 - 57.9iT - 3.48e3T^{2} \)
61 \( 1 + 36.3iT - 3.72e3T^{2} \)
67 \( 1 + 10.0iT - 4.48e3T^{2} \)
71 \( 1 - 64.0iT - 5.04e3T^{2} \)
73 \( 1 - 82.0iT - 5.32e3T^{2} \)
79 \( 1 + 60.4iT - 6.24e3T^{2} \)
83 \( 1 + 112. iT - 6.88e3T^{2} \)
89 \( 1 + 98.9T + 7.92e3T^{2} \)
97 \( 1 + 78.1T + 9.40e3T^{2} \)
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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

−12.25766064950112742428375917838, −11.02006413925184828092630836930, −9.878052970688748667191260852400, −8.884457371581618552519140621157, −8.568820734654613315632704515691, −7.14457772728473706044253897069, −5.88693175805565904157702580638, −4.81925815893799101253444430410, −3.35586566956269622890038587950, −2.53351560961036655352188301031, 0.33183995538229034470347780780, 2.56187585636421504858454680427, 4.04775421402298095654047863552, 4.68756721331721039325718233277, 6.12665297863751728300939660741, 7.60582897693098809275032864329, 8.336370431457319962640525926709, 9.454713731221017118159314200060, 9.799390760223756525396737998042, 11.27092221250282138219406468795

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