Properties

Label 2-2e8-32.5-c3-0-3
Degree $2$
Conductor $256$
Sign $0.923 - 0.382i$
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  + (−3.28 − 7.93i)3-s + (−11.2 − 4.67i)5-s + (11.8 + 11.8i)7-s + (−33.1 + 33.1i)9-s + (−23.5 + 56.9i)11-s + (13.6 − 5.65i)13-s + 104. i·15-s − 44.1i·17-s + (66.5 − 27.5i)19-s + (55.0 − 133. i)21-s + (−60.0 + 60.0i)23-s + (17.1 + 17.1i)25-s + (157. + 65.1i)27-s + (14.3 + 34.6i)29-s + 174.·31-s + ⋯
L(s)  = 1  + (−0.632 − 1.52i)3-s + (−1.00 − 0.418i)5-s + (0.639 + 0.639i)7-s + (−1.22 + 1.22i)9-s + (−0.646 + 1.56i)11-s + (0.291 − 0.120i)13-s + 1.80i·15-s − 0.630i·17-s + (0.803 − 0.332i)19-s + (0.572 − 1.38i)21-s + (−0.544 + 0.544i)23-s + (0.137 + 0.137i)25-s + (1.12 + 0.464i)27-s + (0.0917 + 0.221i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7821955053\)
\(L(\frac12)\) \(\approx\) \(0.7821955053\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.28 + 7.93i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (11.2 + 4.67i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-11.8 - 11.8i)T + 343iT^{2} \)
11 \( 1 + (23.5 - 56.9i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-13.6 + 5.65i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 44.1iT - 4.91e3T^{2} \)
19 \( 1 + (-66.5 + 27.5i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (60.0 - 60.0i)T - 1.21e4iT^{2} \)
29 \( 1 + (-14.3 - 34.6i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 174.T + 2.97e4T^{2} \)
37 \( 1 + (-118. - 49.0i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-15.5 + 15.5i)T - 6.89e4iT^{2} \)
43 \( 1 + (87.3 - 210. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 228. iT - 1.03e5T^{2} \)
53 \( 1 + (258. - 624. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-456. - 188. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (242. + 584. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-332. - 802. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-550. - 550. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-69.2 + 69.2i)T - 3.89e5iT^{2} \)
79 \( 1 - 518. iT - 4.93e5T^{2} \)
83 \( 1 + (595. - 246. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-656. - 656. i)T + 7.04e5iT^{2} \)
97 \( 1 + 388.T + 9.12e5T^{2} \)
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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

−11.85871932945023847200535794350, −11.21535852659550380752260520295, −9.683035548996040019754772094046, −8.198622353337140476308544328258, −7.71291477964013658912122731826, −6.86607785310261005255571142253, −5.52169417838317881086657732365, −4.59042710768062137514280012627, −2.46996476910728667242594220094, −1.13662515331631435490471635181, 0.39797880444826913504086282857, 3.35444719562135337597660138053, 4.04886636866305936886383729829, 5.15183110082357148379795921661, 6.22124436559176329318431096511, 7.81605793271352271724629101633, 8.535480649706295625881549097927, 9.951105626696638669509509326580, 10.74329796220994899713852731916, 11.22639068430742505290809547366

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