Properties

Label 2-2e8-32.13-c3-0-3
Degree $2$
Conductor $256$
Sign $0.382 - 0.923i$
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

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9834712086\)
\(L(\frac12)\) \(\approx\) \(0.9834712086\)
\(L(\frac{5}{2})\) 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

−12.05371496128835253394959151836, −11.12894675426368846700472402689, −9.556369457426883094613906064864, −8.663694576739446160881625248606, −7.61033416645610169296695017805, −7.01133639763084270689621893042, −6.13449570394473753664526140038, −4.13219824146298796502953586503, −2.83671366154475636210856622177, −1.62331643491467693413232682630, 0.35907501563683679000023285598, 3.24370950222171005510632097981, 3.79765106117593053714989609719, 4.75891734735688141072805029556, 6.23903730526306617814427475162, 7.78259358569567656241161126495, 8.756304264181835151565894674801, 9.270541401858249709605733248865, 10.57140625039076841465121941122, 11.03151784982198181184872551219

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