Properties

Label 2-2e8-256.101-c1-0-23
Degree $2$
Conductor $256$
Sign $0.696 - 0.717i$
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  + (0.665 + 1.24i)2-s + (3.13 − 0.786i)3-s + (−1.11 + 1.66i)4-s + (−1.08 − 0.511i)5-s + (3.06 + 3.39i)6-s + (−0.258 − 0.851i)7-s + (−2.81 − 0.286i)8-s + (6.59 − 3.52i)9-s + (−0.0811 − 1.69i)10-s + (1.48 − 0.219i)11-s + (−2.19 + 6.08i)12-s + (−2.17 + 6.07i)13-s + (0.891 − 0.889i)14-s + (−3.80 − 0.756i)15-s + (−1.51 − 3.70i)16-s + (−2.51 + 0.500i)17-s + ⋯
L(s)  = 1  + (0.470 + 0.882i)2-s + (1.81 − 0.454i)3-s + (−0.557 + 0.830i)4-s + (−0.484 − 0.228i)5-s + (1.25 + 1.38i)6-s + (−0.0976 − 0.321i)7-s + (−0.994 − 0.101i)8-s + (2.19 − 1.17i)9-s + (−0.0256 − 0.534i)10-s + (0.446 − 0.0663i)11-s + (−0.633 + 1.75i)12-s + (−0.602 + 1.68i)13-s + (0.238 − 0.237i)14-s + (−0.981 − 0.195i)15-s + (−0.378 − 0.925i)16-s + (−0.609 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10908 + 0.892782i\)
\(L(\frac12)\) \(\approx\) \(2.10908 + 0.892782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.665 - 1.24i)T \)
good3 \( 1 + (-3.13 + 0.786i)T + (2.64 - 1.41i)T^{2} \)
5 \( 1 + (1.08 + 0.511i)T + (3.17 + 3.86i)T^{2} \)
7 \( 1 + (0.258 + 0.851i)T + (-5.82 + 3.88i)T^{2} \)
11 \( 1 + (-1.48 + 0.219i)T + (10.5 - 3.19i)T^{2} \)
13 \( 1 + (2.17 - 6.07i)T + (-10.0 - 8.24i)T^{2} \)
17 \( 1 + (2.51 - 0.500i)T + (15.7 - 6.50i)T^{2} \)
19 \( 1 + (3.15 + 2.85i)T + (1.86 + 18.9i)T^{2} \)
23 \( 1 + (-2.00 + 0.197i)T + (22.5 - 4.48i)T^{2} \)
29 \( 1 + (-0.298 - 0.403i)T + (-8.41 + 27.7i)T^{2} \)
31 \( 1 + (5.73 + 2.37i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.329 + 6.71i)T + (-36.8 - 3.62i)T^{2} \)
41 \( 1 + (6.33 - 7.71i)T + (-7.99 - 40.2i)T^{2} \)
43 \( 1 + (1.69 - 6.75i)T + (-37.9 - 20.2i)T^{2} \)
47 \( 1 + (-7.65 - 5.11i)T + (17.9 + 43.4i)T^{2} \)
53 \( 1 + (-1.28 + 1.73i)T + (-15.3 - 50.7i)T^{2} \)
59 \( 1 + (-0.666 - 1.86i)T + (-45.6 + 37.4i)T^{2} \)
61 \( 1 + (-5.59 + 9.32i)T + (-28.7 - 53.7i)T^{2} \)
67 \( 1 + (-13.3 - 7.98i)T + (31.5 + 59.0i)T^{2} \)
71 \( 1 + (3.21 - 6.00i)T + (-39.4 - 59.0i)T^{2} \)
73 \( 1 + (-0.779 + 2.56i)T + (-60.6 - 40.5i)T^{2} \)
79 \( 1 + (4.86 + 7.27i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-0.695 - 14.1i)T + (-82.6 + 8.13i)T^{2} \)
89 \( 1 + (1.00 + 0.0992i)T + (87.2 + 17.3i)T^{2} \)
97 \( 1 + (-0.850 + 2.05i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.63275053712187739159807469458, −11.53399489737870960396979383841, −9.559735118806380307860435695529, −8.960036726798019474803718385065, −8.195859785560664643462680346038, −7.17095194535064398769756247727, −6.65030923076538109879704482631, −4.43639949179189873356772663606, −3.84753249511263700074063496763, −2.32816302743219020812605360256, 2.15579236877016340699778621931, 3.23318526768576208330561383895, 3.97787827210617178005374963511, 5.30809256855171681022167319548, 7.20391682298231883692640332749, 8.365342555968935549425715828587, 9.043093881863667932490815339963, 10.07318319661818482123327978188, 10.67476578527517273404142018248, 12.10731123687482818772889381635

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