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Properties

Label 2-2e8-32.21-c3-0-8
Degree $2$
Conductor $256$
Sign $0.821 + 0.570i$
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  + (−5.53 + 2.29i)3-s + (−4.22 + 10.1i)5-s + (−11.6 − 11.6i)7-s + (6.27 − 6.27i)9-s + (−27.7 − 11.4i)11-s + (−15.9 − 38.5i)13-s − 66.0i·15-s + 131. i·17-s + (13.7 + 33.3i)19-s + (91.1 + 37.7i)21-s + (128. − 128. i)23-s + (2.32 + 2.32i)25-s + (41.5 − 100. i)27-s + (192. − 79.5i)29-s − 215.·31-s + ⋯
L(s)  = 1  + (−1.06 + 0.441i)3-s + (−0.377 + 0.911i)5-s + (−0.628 − 0.628i)7-s + (0.232 − 0.232i)9-s + (−0.760 − 0.315i)11-s + (−0.340 − 0.822i)13-s − 1.13i·15-s + 1.87i·17-s + (0.166 + 0.402i)19-s + (0.946 + 0.392i)21-s + (1.16 − 1.16i)23-s + (0.0186 + 0.0186i)25-s + (0.296 − 0.715i)27-s + (1.23 − 0.509i)29-s − 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.821 + 0.570i$
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: \((2,\ 256,\ (\ :3/2),\ 0.821 + 0.570i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5951354931\)
\(L(\frac12)\) \(\approx\) \(0.5951354931\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.53 - 2.29i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (4.22 - 10.1i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (11.6 + 11.6i)T + 343iT^{2} \)
11 \( 1 + (27.7 + 11.4i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (15.9 + 38.5i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 131. iT - 4.91e3T^{2} \)
19 \( 1 + (-13.7 - 33.3i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-128. + 128. i)T - 1.21e4iT^{2} \)
29 \( 1 + (-192. + 79.5i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 215.T + 2.97e4T^{2} \)
37 \( 1 + (-9.39 + 22.6i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-257. + 257. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-81.5 - 33.7i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 113. iT - 1.03e5T^{2} \)
53 \( 1 + (-11.8 - 4.92i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (210. - 508. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-251. + 104. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-418. + 173. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (28.0 + 28.0i)T + 3.57e5iT^{2} \)
73 \( 1 + (333. - 333. i)T - 3.89e5iT^{2} \)
79 \( 1 + 38.8iT - 4.93e5T^{2} \)
83 \( 1 + (229. + 555. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (872. + 872. i)T + 7.04e5iT^{2} \)
97 \( 1 + 51.1T + 9.12e5T^{2} \)
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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

−11.09657074502908355046006650386, −10.55995947502190044495554841064, −10.19544867442495327343669749196, −8.464005621362988663518815332479, −7.35561800962460144993600297914, −6.33697192578857982926972392557, −5.45031272988814888397990180223, −4.08745565268843242597598910390, −2.89581711201434382672646873654, −0.37337594383749813067327784543, 0.882184114087245499171046588191, 2.83824665742359077192518887611, 4.81588972432893346173034410604, 5.35662742816713310984982600995, 6.67590384164406897216882182505, 7.48788251484485589915034359754, 8.987499045985830081658514929684, 9.541734046793508232940923565628, 11.09797266645248987998646330452, 11.74817965840546377862205000678

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