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Properties

Label 2-286-11.4-c1-0-3
Degree $2$
Conductor $286$
Sign $0.925 - 0.379i$
Analytic cond. $2.28372$
Root an. cond. $1.51119$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.518 − 1.59i)3-s + (0.309 + 0.951i)4-s + (−2.01 + 1.46i)5-s + (−1.35 + 0.986i)6-s + (1.36 + 4.19i)7-s + (0.309 − 0.951i)8-s + (0.150 + 0.109i)9-s + 2.49·10-s + (0.170 + 3.31i)11-s + 1.67·12-s + (0.809 + 0.587i)13-s + (1.36 − 4.19i)14-s + (1.29 + 3.97i)15-s + (−0.809 + 0.587i)16-s + (1.00 − 0.727i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.299 − 0.921i)3-s + (0.154 + 0.475i)4-s + (−0.901 + 0.655i)5-s + (−0.554 + 0.402i)6-s + (0.515 + 1.58i)7-s + (0.109 − 0.336i)8-s + (0.0500 + 0.0363i)9-s + 0.788·10-s + (0.0512 + 0.998i)11-s + 0.484·12-s + (0.224 + 0.163i)13-s + (0.364 − 1.12i)14-s + (0.333 + 1.02i)15-s + (−0.202 + 0.146i)16-s + (0.242 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(286\)    =    \(2 \cdot 11 \cdot 13\)
Sign: $0.925 - 0.379i$
Analytic conductor: \(2.28372\)
Root analytic conductor: \(1.51119\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{286} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 286,\ (\ :1/2),\ 0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924648 + 0.182265i\)
\(L(\frac12)\) \(\approx\) \(0.924648 + 0.182265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.170 - 3.31i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-0.518 + 1.59i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.01 - 1.46i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.36 - 4.19i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-1.00 + 0.727i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.03 + 3.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.12T + 23T^{2} \)
29 \( 1 + (0.759 + 2.33i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.688 - 0.499i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.97 - 9.15i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.614 - 1.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (0.696 - 2.14i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.26 - 4.55i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.39 + 7.36i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.115 + 0.0835i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.83T + 67T^{2} \)
71 \( 1 + (-4.45 + 3.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.19 + 12.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (13.2 + 9.62i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.06 + 5.86i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + (-8.76 - 6.36i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.95466603969031764355126841065, −11.20965668130896910065548063686, −9.942837792646902278648035943546, −8.887942234480960198552658143575, −7.924680586170464631267300535072, −7.40181469208316904495974484318, −6.22492855570855371217676071229, −4.54621705228621101131368817942, −2.84812035893001560557304664038, −1.87515717451584304963311019696, 0.895753022978011404289578795582, 3.79734361728794323714117996309, 4.23260198534664232421676139030, 5.69135461935086651184961075847, 7.24937443898620107619653139104, 8.029214147240214112474985221295, 8.759701156392788671611531265170, 9.935458533869255087728978175613, 10.60929080093497639964609006616, 11.42210173839360467209379501227

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