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Properties

Label 2-2e8-32.21-c3-0-14
Degree $2$
Conductor $256$
Sign $-0.0137 + 0.999i$
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  + (−1.36 + 0.564i)3-s + (−6.58 + 15.8i)5-s + (−14.5 − 14.5i)7-s + (−17.5 + 17.5i)9-s + (34.4 + 14.2i)11-s + (−15.3 − 37.1i)13-s − 25.3i·15-s − 103. i·17-s + (12.4 + 29.9i)19-s + (28.0 + 11.6i)21-s + (72.3 − 72.3i)23-s + (−120. − 120. i)25-s + (29.2 − 70.5i)27-s + (−23.9 + 9.90i)29-s + 124.·31-s + ⋯
L(s)  = 1  + (−0.262 + 0.108i)3-s + (−0.588 + 1.42i)5-s + (−0.785 − 0.785i)7-s + (−0.650 + 0.650i)9-s + (0.944 + 0.391i)11-s + (−0.328 − 0.792i)13-s − 0.436i·15-s − 1.47i·17-s + (0.149 + 0.361i)19-s + (0.291 + 0.120i)21-s + (0.656 − 0.656i)23-s + (−0.966 − 0.966i)25-s + (0.208 − 0.503i)27-s + (−0.153 + 0.0634i)29-s + 0.722·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.0137 + 0.999i$
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: \((2,\ 256,\ (\ :3/2),\ -0.0137 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5399642148\)
\(L(\frac12)\) \(\approx\) \(0.5399642148\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.36 - 0.564i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (6.58 - 15.8i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (14.5 + 14.5i)T + 343iT^{2} \)
11 \( 1 + (-34.4 - 14.2i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (15.3 + 37.1i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 103. iT - 4.91e3T^{2} \)
19 \( 1 + (-12.4 - 29.9i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-72.3 + 72.3i)T - 1.21e4iT^{2} \)
29 \( 1 + (23.9 - 9.90i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + (18.0 - 43.5i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (45.1 - 45.1i)T - 6.89e4iT^{2} \)
43 \( 1 + (457. + 189. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 582. iT - 1.03e5T^{2} \)
53 \( 1 + (395. + 163. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-142. + 344. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-34.8 + 14.4i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-196. + 81.4i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-520. - 520. i)T + 3.57e5iT^{2} \)
73 \( 1 + (582. - 582. i)T - 3.89e5iT^{2} \)
79 \( 1 - 157. iT - 4.93e5T^{2} \)
83 \( 1 + (-54.5 - 131. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (272. + 272. i)T + 7.04e5iT^{2} \)
97 \( 1 - 788.T + 9.12e5T^{2} \)
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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

−11.31861178296047998261844505165, −10.39448118191105496503079549697, −9.774752266753040654539830341115, −8.225638471473569985163334795666, −7.06678728057298531350002694836, −6.66740521620587494967292096367, −5.10380191818635008848618078322, −3.64041237070900405402686053520, −2.74361904569732367069804346448, −0.23923052465975926083169100006, 1.28044879068974712768139513693, 3.33859739100265925725637421321, 4.52499063068772035928589227755, 5.79940296517576215513061184036, 6.58916724368983900579591039223, 8.191197719039871260350836328304, 9.022433725063781948134367063535, 9.454356257174721449308584068023, 11.21712780365306728562460860960, 12.01430582133395828367109997436

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