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Properties

Label 2-255-85.22-c1-0-17
Degree $2$
Conductor $255$
Sign $-0.986 + 0.163i$
Analytic cond. $2.03618$
Root an. cond. $1.42694$
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  + (1.03 − 2.49i)2-s + (0.980 + 0.195i)3-s + (−3.74 − 3.74i)4-s + (−1.65 − 1.49i)5-s + (1.50 − 2.24i)6-s + (−0.128 + 0.192i)7-s + (−8.20 + 3.39i)8-s + (0.923 + 0.382i)9-s + (−5.45 + 2.58i)10-s + (1.99 + 1.33i)11-s + (−2.93 − 4.39i)12-s + 4.63·13-s + (0.347 + 0.520i)14-s + (−1.33 − 1.79i)15-s + 13.4i·16-s + (1.87 − 3.67i)17-s + ⋯
L(s)  = 1  + (0.730 − 1.76i)2-s + (0.566 + 0.112i)3-s + (−1.87 − 1.87i)4-s + (−0.741 − 0.670i)5-s + (0.612 − 0.916i)6-s + (−0.0486 + 0.0728i)7-s + (−2.90 + 1.20i)8-s + (0.307 + 0.127i)9-s + (−1.72 + 0.818i)10-s + (0.602 + 0.402i)11-s + (−0.848 − 1.26i)12-s + 1.28·13-s + (0.0929 + 0.139i)14-s + (−0.344 − 0.463i)15-s + 3.35i·16-s + (0.454 − 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 255 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(255\)    =    \(3 \cdot 5 \cdot 17\)
Sign: $-0.986 + 0.163i$
Analytic conductor: \(2.03618\)
Root analytic conductor: \(1.42694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{255} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 255,\ (\ :1/2),\ -0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139219 - 1.69652i\)
\(L(\frac12)\) \(\approx\) \(0.139219 - 1.69652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.980 - 0.195i)T \)
5 \( 1 + (1.65 + 1.49i)T \)
17 \( 1 + (-1.87 + 3.67i)T \)
good2 \( 1 + (-1.03 + 2.49i)T + (-1.41 - 1.41i)T^{2} \)
7 \( 1 + (0.128 - 0.192i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.99 - 1.33i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 - 4.63T + 13T^{2} \)
19 \( 1 + (3.42 - 1.41i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.825 + 4.15i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-1.70 + 8.58i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-2.18 + 1.45i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.72 - 8.66i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.945 - 4.75i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.153 - 0.371i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + (-10.6 - 4.42i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.594 - 1.43i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.69 - 0.933i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-1.69 + 1.69i)T - 67iT^{2} \)
71 \( 1 + (8.33 + 12.4i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-0.341 - 0.510i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (1.83 - 2.75i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (4.21 - 10.1i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-1.93 - 1.93i)T + 89iT^{2} \)
97 \( 1 + (1.26 + 1.89i)T + (-37.1 + 89.6i)T^{2} \)
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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

−11.78140579931647210614953259262, −10.85051451579374516354845780288, −9.824802271585752426071475264094, −8.996850952667087883911974337856, −8.127359063160666095136887261090, −6.11385272891704163522623041714, −4.57923159125623144576734649157, −4.03303322555315011114518658962, −2.81387507174722883404981705193, −1.20352928747203489487322414660, 3.51395891896878251670828774375, 3.97197053072534332516860268181, 5.62717205932900722136973227261, 6.63874989664801478593768094265, 7.30707941036302878915476979288, 8.459643270416048873281488540989, 8.789493322850427982964102975507, 10.54863274269918174655903824640, 11.88418077873580671158179440197, 12.89322149004226628761268392814

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