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Properties

Label 2-2e8-32.21-c3-0-10
Degree $2$
Conductor $256$
Sign $0.998 + 0.0479i$
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  + (−6.06 + 2.51i)3-s + (2.91 − 7.02i)5-s + (13.3 + 13.3i)7-s + (11.3 − 11.3i)9-s + (−49.6 − 20.5i)11-s + (−8.74 − 21.1i)13-s + 49.9i·15-s − 77.7i·17-s + (53.3 + 128. i)19-s + (−114. − 47.5i)21-s + (35.5 − 35.5i)23-s + (47.4 + 47.4i)25-s + (27.3 − 66.0i)27-s + (245. − 101. i)29-s + 202.·31-s + ⋯
L(s)  = 1  + (−1.16 + 0.483i)3-s + (0.260 − 0.628i)5-s + (0.722 + 0.722i)7-s + (0.421 − 0.421i)9-s + (−1.36 − 0.563i)11-s + (−0.186 − 0.450i)13-s + 0.859i·15-s − 1.10i·17-s + (0.643 + 1.55i)19-s + (−1.19 − 0.494i)21-s + (0.321 − 0.321i)23-s + (0.379 + 0.379i)25-s + (0.195 − 0.470i)27-s + (1.57 − 0.650i)29-s + 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0479i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.998 + 0.0479i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.998 + 0.0479i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.175514783\)
\(L(\frac12)\) \(\approx\) \(1.175514783\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.06 - 2.51i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-2.91 + 7.02i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-13.3 - 13.3i)T + 343iT^{2} \)
11 \( 1 + (49.6 + 20.5i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (8.74 + 21.1i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 77.7iT - 4.91e3T^{2} \)
19 \( 1 + (-53.3 - 128. i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-35.5 + 35.5i)T - 1.21e4iT^{2} \)
29 \( 1 + (-245. + 101. i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 + (36.3 - 87.6i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (36.8 - 36.8i)T - 6.89e4iT^{2} \)
43 \( 1 + (-185. - 76.8i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 82.9iT - 1.03e5T^{2} \)
53 \( 1 + (-534. - 221. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-75.7 + 182. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-472. + 195. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (102. - 42.5i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (520. + 520. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-244. + 244. i)T - 3.89e5iT^{2} \)
79 \( 1 + 774. iT - 4.93e5T^{2} \)
83 \( 1 + (23.9 + 57.7i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (351. + 351. i)T + 7.04e5iT^{2} \)
97 \( 1 - 1.30e3T + 9.12e5T^{2} \)
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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

−11.69513347140742476741299211539, −10.59087668863466601566725088967, −9.952910212720480666475492232592, −8.597717449493482702886600814380, −7.80111382177266396228141677745, −6.07943458621196429664911347885, −5.25760085769870146032906785075, −4.81236065419575648242603548604, −2.73379516722051651929094677643, −0.76615318736790892114671529249, 0.919985818065346945971035796474, 2.59345550928979233949622533100, 4.56965767578214125994204409428, 5.43306667221269537537430807462, 6.73077321831575759749749994766, 7.25297213628793245094154274766, 8.504329748981060932616202277204, 10.16387951621674655994608872214, 10.69499436135047603090688601153, 11.46291174246955153880010471424

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