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Properties

Label 2-2e8-32.21-c3-0-15
Degree $2$
Conductor $256$
Sign $0.450 + 0.892i$
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  + (−2.66 + 1.10i)3-s + (5.50 − 13.2i)5-s + (6.48 + 6.48i)7-s + (−13.2 + 13.2i)9-s + (49.3 + 20.4i)11-s + (−21.4 − 51.7i)13-s + 41.4i·15-s + 3.73i·17-s + (−36.7 − 88.6i)19-s + (−24.4 − 10.1i)21-s + (45.4 − 45.4i)23-s + (−58.0 − 58.0i)25-s + (50.3 − 121. i)27-s + (−51.9 + 21.5i)29-s + 73.5·31-s + ⋯
L(s)  = 1  + (−0.512 + 0.212i)3-s + (0.492 − 1.18i)5-s + (0.349 + 0.349i)7-s + (−0.489 + 0.489i)9-s + (1.35 + 0.560i)11-s + (−0.457 − 1.10i)13-s + 0.714i·15-s + 0.0533i·17-s + (−0.443 − 1.07i)19-s + (−0.253 − 0.105i)21-s + (0.412 − 0.412i)23-s + (−0.464 − 0.464i)25-s + (0.359 − 0.867i)27-s + (−0.332 + 0.137i)29-s + 0.425·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.450 + 0.892i$
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: \((2,\ 256,\ (\ :3/2),\ 0.450 + 0.892i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.564226487\)
\(L(\frac12)\) \(\approx\) \(1.564226487\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.66 - 1.10i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-5.50 + 13.2i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-6.48 - 6.48i)T + 343iT^{2} \)
11 \( 1 + (-49.3 - 20.4i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (21.4 + 51.7i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 3.73iT - 4.91e3T^{2} \)
19 \( 1 + (36.7 + 88.6i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-45.4 + 45.4i)T - 1.21e4iT^{2} \)
29 \( 1 + (51.9 - 21.5i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 73.5T + 2.97e4T^{2} \)
37 \( 1 + (-165. + 399. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-334. + 334. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-328. - 136. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 185. iT - 1.03e5T^{2} \)
53 \( 1 + (412. + 171. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (214. - 518. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (85.1 - 35.2i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (252. - 104. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (430. + 430. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-41.8 + 41.8i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.21e3iT - 4.93e5T^{2} \)
83 \( 1 + (290. + 702. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-365. - 365. i)T + 7.04e5iT^{2} \)
97 \( 1 - 508.T + 9.12e5T^{2} \)
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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

−11.43897298202369763599690152847, −10.53455742711176838247116180386, −9.278471822532456696371594178262, −8.778841819578170251700727193279, −7.50742997592744030937659877554, −6.04320969944836169435597688159, −5.19834169137403672685100095377, −4.37040663411854780882876996097, −2.31480220117366852941408732873, −0.72622701983175321790122340174, 1.38943824108587811059497194885, 3.04500808078470234133901191893, 4.35634377731581433836851812341, 6.18047246666817005215251120133, 6.38434376245727367840940994102, 7.58578397107116917701568533034, 9.017387754235406954770580048119, 9.863810056964360683035303591988, 11.10192821198974894443251317005, 11.47241430183522821544736328111

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