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Properties

Label 2-2e8-8.5-c3-0-2
Degree $2$
Conductor $256$
Sign $-0.707 + 0.707i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.92i·3-s + 11.8i·5-s − 9.85·7-s − 52.7·9-s − 39.0i·11-s + 91.5i·13-s − 105.·15-s − 37.1·17-s − 46.4i·19-s − 87.9i·21-s + 120.·23-s − 15.5·25-s − 229. i·27-s − 27.2i·29-s − 81.1·31-s + ⋯
L(s)  = 1  + 1.71i·3-s + 1.06i·5-s − 0.532·7-s − 1.95·9-s − 1.07i·11-s + 1.95i·13-s − 1.82·15-s − 0.529·17-s − 0.561i·19-s − 0.914i·21-s + 1.09·23-s − 0.124·25-s − 1.63i·27-s − 0.174i·29-s − 0.470·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9769915141\)
\(L(\frac12)\) \(\approx\) \(0.9769915141\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 8.92iT - 27T^{2} \)
5 \( 1 - 11.8iT - 125T^{2} \)
7 \( 1 + 9.85T + 343T^{2} \)
11 \( 1 + 39.0iT - 1.33e3T^{2} \)
13 \( 1 - 91.5iT - 2.19e3T^{2} \)
17 \( 1 + 37.1T + 4.91e3T^{2} \)
19 \( 1 + 46.4iT - 6.85e3T^{2} \)
23 \( 1 - 120.T + 1.21e4T^{2} \)
29 \( 1 + 27.2iT - 2.43e4T^{2} \)
31 \( 1 + 81.1T + 2.97e4T^{2} \)
37 \( 1 + 10.9iT - 5.06e4T^{2} \)
41 \( 1 - 205.T + 6.89e4T^{2} \)
43 \( 1 + 115. iT - 7.95e4T^{2} \)
47 \( 1 + 312.T + 1.03e5T^{2} \)
53 \( 1 + 90.9iT - 1.48e5T^{2} \)
59 \( 1 - 550. iT - 2.05e5T^{2} \)
61 \( 1 - 630. iT - 2.26e5T^{2} \)
67 \( 1 + 661. iT - 3.00e5T^{2} \)
71 \( 1 + 494.T + 3.57e5T^{2} \)
73 \( 1 + 566.T + 3.89e5T^{2} \)
79 \( 1 - 49.4T + 4.93e5T^{2} \)
83 \( 1 - 564. iT - 5.71e5T^{2} \)
89 \( 1 + 1.08e3T + 7.04e5T^{2} \)
97 \( 1 - 464.T + 9.12e5T^{2} \)
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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

−11.54961635874440644154560944127, −11.12170575513587133309082460585, −10.34744764031332425552784638705, −9.272204289654731492686572728026, −8.847291522299253767329951397604, −7.00325267386804052595945813729, −6.10788403549982966185195191672, −4.72739685823402386405169901815, −3.71002657950735180142754015905, −2.76934359926576829859527878266, 0.38486846916063418233492099180, 1.53656274133624747750135255868, 2.97898206473840872037561366224, 4.96098072687574196728508073086, 5.99866826119698498683691037349, 7.10542638423476562296446841520, 7.909620961497077057998670253811, 8.743985203005158327860411077041, 9.926124212835015426504679428244, 11.24963210040846828082256660327

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