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Properties

Label 2-2e8-32.13-c3-0-14
Degree $2$
Conductor $256$
Sign $0.708 + 0.705i$
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  + (1.64 − 3.96i)3-s + (11.8 − 4.89i)5-s + (5.11 − 5.11i)7-s + (6.09 + 6.09i)9-s + (15.2 + 36.8i)11-s + (73.4 + 30.4i)13-s − 54.7i·15-s − 66.8i·17-s + (−37.0 − 15.3i)19-s + (−11.8 − 28.6i)21-s + (−30.1 − 30.1i)23-s + (27.1 − 27.1i)25-s + (141. − 58.4i)27-s + (−64.4 + 155. i)29-s − 219.·31-s + ⋯
L(s)  = 1  + (0.315 − 0.762i)3-s + (1.05 − 0.437i)5-s + (0.276 − 0.276i)7-s + (0.225 + 0.225i)9-s + (0.417 + 1.00i)11-s + (1.56 + 0.649i)13-s − 0.943i·15-s − 0.954i·17-s + (−0.447 − 0.185i)19-s + (−0.123 − 0.297i)21-s + (−0.273 − 0.273i)23-s + (0.217 − 0.217i)25-s + (1.00 − 0.416i)27-s + (−0.412 + 0.996i)29-s − 1.26·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.708 + 0.705i$
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: \((2,\ 256,\ (\ :3/2),\ 0.708 + 0.705i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.720460862\)
\(L(\frac12)\) \(\approx\) \(2.720460862\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.64 + 3.96i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-11.8 + 4.89i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-5.11 + 5.11i)T - 343iT^{2} \)
11 \( 1 + (-15.2 - 36.8i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (-73.4 - 30.4i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 66.8iT - 4.91e3T^{2} \)
19 \( 1 + (37.0 + 15.3i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (30.1 + 30.1i)T + 1.21e4iT^{2} \)
29 \( 1 + (64.4 - 155. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 219.T + 2.97e4T^{2} \)
37 \( 1 + (-286. + 118. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-64.2 - 64.2i)T + 6.89e4iT^{2} \)
43 \( 1 + (200. + 484. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 392. iT - 1.03e5T^{2} \)
53 \( 1 + (107. + 258. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-237. + 98.4i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-43.9 + 106. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (333. - 804. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (387. - 387. i)T - 3.57e5iT^{2} \)
73 \( 1 + (518. + 518. i)T + 3.89e5iT^{2} \)
79 \( 1 - 214. iT - 4.93e5T^{2} \)
83 \( 1 + (-436. - 180. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (877. - 877. i)T - 7.04e5iT^{2} \)
97 \( 1 - 43.7T + 9.12e5T^{2} \)
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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

−11.51351863461272600754785744977, −10.45822163700412066304238164853, −9.372949462477144446328957132237, −8.650395310919842885186714146816, −7.35976799248711902553906767878, −6.60211443015724363367966388582, −5.35156180490950161188974538639, −4.09089206117311116430829960537, −2.11438963259067778886072738865, −1.34404790718078502987134769774, 1.48012463040575653901071528406, 3.15068701388349611196966859203, 4.14215601041995360814604948432, 5.89889196964150409154112595684, 6.20404447539364674532635110190, 8.063004877919307354005932895024, 8.924251432857911375081449370748, 9.790650838471533500763586020568, 10.66825797615342142914449550697, 11.34792861461201584467432243587

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