Properties

Label 2-2e8-32.13-c1-0-4
Degree $2$
Conductor $256$
Sign $0.555 + 0.831i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
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  + (0.707 − 1.70i)3-s + (3.12 − 1.29i)5-s + (−1 + i)7-s + (−0.292 − 0.292i)9-s + (0.121 + 0.292i)11-s + (−1.70 − 0.707i)13-s − 6.24i·15-s + 2.82i·17-s + (−5.53 − 2.29i)19-s + (0.999 + 2.41i)21-s + (−0.171 − 0.171i)23-s + (4.53 − 4.53i)25-s + (4.41 − 1.82i)27-s + (−1.12 + 2.70i)29-s + 4·31-s + ⋯
L(s)  = 1  + (0.408 − 0.985i)3-s + (1.39 − 0.578i)5-s + (−0.377 + 0.377i)7-s + (−0.0976 − 0.0976i)9-s + (0.0365 + 0.0883i)11-s + (−0.473 − 0.196i)13-s − 1.61i·15-s + 0.685i·17-s + (−1.26 − 0.526i)19-s + (0.218 + 0.526i)21-s + (−0.0357 − 0.0357i)23-s + (0.907 − 0.907i)25-s + (0.849 − 0.351i)27-s + (−0.208 + 0.502i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45245 - 0.776355i\)
\(L(\frac12)\) \(\approx\) \(1.45245 - 0.776355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.707 + 1.70i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-3.12 + 1.29i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-0.121 - 0.292i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.70 + 0.707i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + (5.53 + 2.29i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.171 + 0.171i)T + 23iT^{2} \)
29 \( 1 + (1.12 - 2.70i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (1.70 - 0.707i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.82 + 5.82i)T + 41iT^{2} \)
43 \( 1 + (-3.29 - 7.94i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (3.12 + 7.53i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.12 - 2.53i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.292 - 0.707i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.53 + 3.70i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-0.171 + 0.171i)T - 71iT^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-6.12 - 2.53i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.65 - 2.65i)T - 89iT^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
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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

−12.48111179317740827082437101154, −10.79514573516505097937504853930, −9.806194492846366324785121846294, −8.940556600901434289952505883115, −8.046690377839520544090238194354, −6.73647444510445565589765587313, −5.99248497263894253253794658931, −4.73713303565563725836505926758, −2.63129916045971228290448228239, −1.62456651384583248660789909657, 2.27388619249460429245900498453, 3.57789961753940801480259924595, 4.84931891560977910494127252159, 6.14019827252509520796946122644, 7.01486095300628755445851119949, 8.612489646639570121176308646373, 9.619539174654769335542201518326, 10.06998874607469122729459827728, 10.77453010729469106516727179332, 12.18702740014711959242028943035

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