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Properties

Label 2-287-287.286-c2-0-41
Degree $2$
Conductor $287$
Sign $-0.915 + 0.401i$
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 + 0.913·3-s − 3·4-s − 7.47i·5-s − 0.913·6-s + (6.10 + 3.41i)7-s + 7·8-s − 8.16·9-s + 7.47i·10-s − 6.11i·11-s − 2.74·12-s − 0.913·13-s + (−6.10 − 3.41i)14-s − 6.83i·15-s + 5·16-s + 1.10·17-s + ⋯
L(s)  = 1  − 0.5·2-s + 0.304·3-s − 0.750·4-s − 1.49i·5-s − 0.152·6-s + (0.872 + 0.488i)7-s + 0.875·8-s − 0.907·9-s + 0.747i·10-s − 0.556i·11-s − 0.228·12-s − 0.0702·13-s + (−0.436 − 0.244i)14-s − 0.455i·15-s + 0.312·16-s + 0.0649·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.915 + 0.401i$
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: \((2,\ 287,\ (\ :1),\ -0.915 + 0.401i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.116810 - 0.556699i\)
\(L(\frac12)\) \(\approx\) \(0.116810 - 0.556699i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-6.10 - 3.41i)T \)
41 \( 1 + (-24.7 + 32.7i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 - 0.913T + 9T^{2} \)
5 \( 1 + 7.47iT - 25T^{2} \)
11 \( 1 + 6.11iT - 121T^{2} \)
13 \( 1 + 0.913T + 169T^{2} \)
17 \( 1 - 1.10T + 289T^{2} \)
19 \( 1 + 28.6T + 361T^{2} \)
23 \( 1 + 26.7T + 529T^{2} \)
29 \( 1 + 52.5iT - 841T^{2} \)
31 \( 1 - 26.7iT - 961T^{2} \)
37 \( 1 + 18.8T + 1.36e3T^{2} \)
43 \( 1 + 19.4T + 1.84e3T^{2} \)
47 \( 1 + 48.5T + 2.20e3T^{2} \)
53 \( 1 + 44.2iT - 2.80e3T^{2} \)
59 \( 1 + 19.3iT - 3.48e3T^{2} \)
61 \( 1 + 28.0iT - 3.72e3T^{2} \)
67 \( 1 + 14.3iT - 4.48e3T^{2} \)
71 \( 1 - 134. iT - 5.04e3T^{2} \)
73 \( 1 - 44.8iT - 5.32e3T^{2} \)
79 \( 1 + 70.8iT - 6.24e3T^{2} \)
83 \( 1 - 90.3iT - 6.88e3T^{2} \)
89 \( 1 - 81.5T + 7.92e3T^{2} \)
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

−11.22277863988870411798644468747, −9.963264436086793351367470729150, −8.932593561014700029587824165577, −8.404052757479211100248092928106, −8.049462555759550985459140390116, −5.89644699054034246632844493609, −5.00348693390152203898279689786, −4.05328211309039808459537132031, −1.93720070913977626587859657954, −0.31845898052079239604538409063, 2.05677782839799528756720172647, 3.56118189773721353827945861448, 4.74219086710803772735205274633, 6.22781376514348479886467323673, 7.43161947318866037662654494311, 8.136589915260370408365328393369, 9.099144822441238694312781954663, 10.32933554380339246407215425941, 10.71711410813976009745654610391, 11.73295145103046452279095970709

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