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Properties

Label 2-287-287.286-c2-0-23
Degree $2$
Conductor $287$
Sign $0.0929 - 0.995i$
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  + 1.16·2-s + 4.43·3-s − 2.63·4-s + 9.05i·5-s + 5.17·6-s + (−6.69 − 2.05i)7-s − 7.74·8-s + 10.6·9-s + 10.5i·10-s + 15.2i·11-s − 11.6·12-s + 11.9·13-s + (−7.80 − 2.40i)14-s + 40.1i·15-s + 1.50·16-s + 17.3·17-s + ⋯
L(s)  = 1  + 0.583·2-s + 1.47·3-s − 0.659·4-s + 1.81i·5-s + 0.862·6-s + (−0.955 − 0.294i)7-s − 0.968·8-s + 1.18·9-s + 1.05i·10-s + 1.38i·11-s − 0.974·12-s + 0.918·13-s + (−0.557 − 0.171i)14-s + 2.67i·15-s + 0.0942·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0929 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0929 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0929 - 0.995i$
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: \((2,\ 287,\ (\ :1),\ 0.0929 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.87526 + 1.70844i\)
\(L(\frac12)\) \(\approx\) \(1.87526 + 1.70844i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (6.69 + 2.05i)T \)
41 \( 1 + (8.37 + 40.1i)T \)
good2 \( 1 - 1.16T + 4T^{2} \)
3 \( 1 - 4.43T + 9T^{2} \)
5 \( 1 - 9.05iT - 25T^{2} \)
11 \( 1 - 15.2iT - 121T^{2} \)
13 \( 1 - 11.9T + 169T^{2} \)
17 \( 1 - 17.3T + 289T^{2} \)
19 \( 1 + 0.333T + 361T^{2} \)
23 \( 1 - 14.2T + 529T^{2} \)
29 \( 1 - 9.15iT - 841T^{2} \)
31 \( 1 + 50.9iT - 961T^{2} \)
37 \( 1 - 16.3T + 1.36e3T^{2} \)
43 \( 1 + 38.9T + 1.84e3T^{2} \)
47 \( 1 - 79.8T + 2.20e3T^{2} \)
53 \( 1 - 28.3iT - 2.80e3T^{2} \)
59 \( 1 - 89.3iT - 3.48e3T^{2} \)
61 \( 1 + 35.2iT - 3.72e3T^{2} \)
67 \( 1 + 62.8iT - 4.48e3T^{2} \)
71 \( 1 + 30.5iT - 5.04e3T^{2} \)
73 \( 1 + 19.9iT - 5.32e3T^{2} \)
79 \( 1 - 38.5iT - 6.24e3T^{2} \)
83 \( 1 - 113. iT - 6.88e3T^{2} \)
89 \( 1 - 141.T + 7.92e3T^{2} \)
97 \( 1 - 86.9T + 9.40e3T^{2} \)
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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

−12.08683677294884668278051706086, −10.59401224718806628164595234893, −9.803978483335115390639996912490, −9.176663915566614695718191881649, −7.80157485990849006232129115290, −7.04782885311132607429358355629, −5.95472679223736670008988612206, −4.01439329219870447837794908239, −3.39684071112299285351375818232, −2.50457553216178347050822090835, 0.960946123039635297438173420901, 3.15812203993735106130309733605, 3.79711230663729460301278021485, 5.13749122297945276196013674557, 6.05697709701860845209802983339, 8.045755628357790386963766436157, 8.785964324084758896141906074433, 8.975256757526385874045409087781, 9.974417435639909739593983130260, 11.83840786685110190552751119526

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