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Properties

Label 2-2e8-32.13-c3-0-17
Degree $2$
Conductor $256$
Sign $0.646 + 0.762i$
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  + (−0.477 + 1.15i)3-s + (16.3 − 6.77i)5-s + (18.0 − 18.0i)7-s + (17.9 + 17.9i)9-s + (−20.1 − 48.5i)11-s + (−37.8 − 15.6i)13-s + 22.0i·15-s + 53.0i·17-s + (−32.4 − 13.4i)19-s + (12.1 + 29.4i)21-s + (−32.1 − 32.1i)23-s + (132. − 132. i)25-s + (−60.4 + 25.0i)27-s + (52.0 − 125. i)29-s + 53.3·31-s + ⋯
L(s)  = 1  + (−0.0919 + 0.221i)3-s + (1.46 − 0.605i)5-s + (0.973 − 0.973i)7-s + (0.666 + 0.666i)9-s + (−0.551 − 1.33i)11-s + (−0.806 − 0.334i)13-s + 0.380i·15-s + 0.757i·17-s + (−0.391 − 0.162i)19-s + (0.126 + 0.305i)21-s + (−0.291 − 0.291i)23-s + (1.06 − 1.06i)25-s + (−0.431 + 0.178i)27-s + (0.333 − 0.804i)29-s + 0.309·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.646 + 0.762i$
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: \((2,\ 256,\ (\ :3/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.407956587\)
\(L(\frac12)\) \(\approx\) \(2.407956587\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.477 - 1.15i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-16.3 + 6.77i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-18.0 + 18.0i)T - 343iT^{2} \)
11 \( 1 + (20.1 + 48.5i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (37.8 + 15.6i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 53.0iT - 4.91e3T^{2} \)
19 \( 1 + (32.4 + 13.4i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (32.1 + 32.1i)T + 1.21e4iT^{2} \)
29 \( 1 + (-52.0 + 125. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 53.3T + 2.97e4T^{2} \)
37 \( 1 + (-57.3 + 23.7i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-240. - 240. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-56.3 - 135. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 314. iT - 1.03e5T^{2} \)
53 \( 1 + (-177. - 428. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (133. - 55.4i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-191. + 462. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (55.4 - 133. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (191. - 191. i)T - 3.57e5iT^{2} \)
73 \( 1 + (175. + 175. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.22e3iT - 4.93e5T^{2} \)
83 \( 1 + (-896. - 371. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-883. + 883. i)T - 7.04e5iT^{2} \)
97 \( 1 + 682.T + 9.12e5T^{2} \)
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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

−11.10683507067167512895259832272, −10.44598192661434935129562472445, −9.768036595088392609887722277148, −8.454484580858276317849113959532, −7.68144642336347389567982381715, −6.17183100385619182611709337892, −5.20658776799156767738321017616, −4.33450481388045510566952160124, −2.32020301829357096588071212017, −1.02813005366418199058764045413, 1.77622340663089200367352144531, 2.49436373212186753982304482471, 4.67127477270338894476538585695, 5.58544720141847662869237921994, 6.71562021440592733107199731638, 7.56112460454675821558715747151, 9.102262808946615105187984703259, 9.759583189555997311366043977978, 10.57580213927983148524775792812, 11.90881483585298886911340811830

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