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Properties

Label 2-255-255.74-c1-0-14
Degree $2$
Conductor $255$
Sign $0.189 + 0.981i$
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  + (−2.33 − 0.968i)2-s + (1.69 − 0.337i)3-s + (3.11 + 3.11i)4-s + (1.24 − 1.85i)5-s + (−4.29 − 0.855i)6-s + (−2.33 − 5.62i)8-s + (2.77 − 1.14i)9-s + (−4.70 + 3.14i)10-s + (6.34 + 4.23i)12-s + (1.48 − 3.57i)15-s + 6.60i·16-s + (2.86 + 2.96i)17-s − 7.59·18-s + (5.19 + 2.15i)19-s + (9.66 − 1.92i)20-s + ⋯
L(s)  = 1  + (−1.65 − 0.684i)2-s + (0.980 − 0.195i)3-s + (1.55 + 1.55i)4-s + (0.555 − 0.831i)5-s + (−1.75 − 0.349i)6-s + (−0.823 − 1.98i)8-s + (0.923 − 0.382i)9-s + (−1.48 + 0.994i)10-s + (1.83 + 1.22i)12-s + (0.382 − 0.923i)15-s + 1.65i·16-s + (0.695 + 0.718i)17-s − 1.78·18-s + (1.19 + 0.493i)19-s + (2.16 − 0.429i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 255 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(255\)    =    \(3 \cdot 5 \cdot 17\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(2.03618\)
Root analytic conductor: \(1.42694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{255} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 255,\ (\ :1/2),\ 0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.700177 - 0.577987i\)
\(L(\frac12)\) \(\approx\) \(0.700177 - 0.577987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 + 0.337i)T \)
5 \( 1 + (-1.24 + 1.85i)T \)
17 \( 1 + (-2.86 - 2.96i)T \)
good2 \( 1 + (2.33 + 0.968i)T + (1.41 + 1.41i)T^{2} \)
7 \( 1 + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 13iT^{2} \)
19 \( 1 + (-5.19 - 2.15i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (8.60 + 1.71i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (2.13 + 10.7i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (2.59 - 2.59i)T - 47iT^{2} \)
53 \( 1 + (-11.7 - 4.86i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (9.78 - 6.53i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (2.99 - 15.0i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.63 + 6.36i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (-37.1 - 89.6i)T^{2} \)
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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

−11.81854051362404167491521680782, −10.32484744256214877239420523355, −9.748375522103339185509898750450, −9.073253608132194169207536825652, −8.080356271047670009715347243301, −7.64129986046065694542621378644, −6.00624215184559699730803258022, −3.90056582140863514735417495816, −2.39405042106107616708688811255, −1.28659412493656296807813632182, 1.78487125048311643411766715116, 3.20068566775361034605215466363, 5.46820835940385068232692689207, 6.82122995521782966517797434323, 7.46197213454167659398759703829, 8.374293265626007070779108754033, 9.421940038494671859338978060466, 9.919437148015713927274243397028, 10.65895973907772981395464059819, 11.88377486044246183826291019394

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