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Properties

Label 2-2e8-32.29-c3-0-10
Degree $2$
Conductor $256$
Sign $0.123 - 0.992i$
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  + (7.57 + 3.13i)3-s + (8.03 + 19.4i)5-s + (1.85 − 1.85i)7-s + (28.3 + 28.3i)9-s + (8.63 − 3.57i)11-s + (11.5 − 27.9i)13-s + 172. i·15-s − 7.99i·17-s + (5.76 − 13.9i)19-s + (19.8 − 8.20i)21-s + (−60.2 − 60.2i)23-s + (−223. + 223. i)25-s + (41.2 + 99.4i)27-s + (−167. − 69.3i)29-s + 225.·31-s + ⋯
L(s)  = 1  + (1.45 + 0.603i)3-s + (0.718 + 1.73i)5-s + (0.0999 − 0.0999i)7-s + (1.05 + 1.05i)9-s + (0.236 − 0.0979i)11-s + (0.247 − 0.597i)13-s + 2.96i·15-s − 0.114i·17-s + (0.0695 − 0.167i)19-s + (0.205 − 0.0853i)21-s + (−0.546 − 0.546i)23-s + (−1.78 + 1.78i)25-s + (0.293 + 0.709i)27-s + (−1.07 − 0.444i)29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.123 - 0.992i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.123 - 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.343644379\)
\(L(\frac12)\) \(\approx\) \(3.343644379\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.57 - 3.13i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-8.03 - 19.4i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-1.85 + 1.85i)T - 343iT^{2} \)
11 \( 1 + (-8.63 + 3.57i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-11.5 + 27.9i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 7.99iT - 4.91e3T^{2} \)
19 \( 1 + (-5.76 + 13.9i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (60.2 + 60.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (167. + 69.3i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 + (0.431 + 1.04i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (275. + 275. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-257. + 106. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 51.3iT - 1.03e5T^{2} \)
53 \( 1 + (-2.98 + 1.23i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-101. - 244. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-270. - 111. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-778. - 322. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (484. - 484. i)T - 3.57e5iT^{2} \)
73 \( 1 + (212. + 212. i)T + 3.89e5iT^{2} \)
79 \( 1 - 593. iT - 4.93e5T^{2} \)
83 \( 1 + (-320. + 773. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-435. + 435. i)T - 7.04e5iT^{2} \)
97 \( 1 + 570.T + 9.12e5T^{2} \)
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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

−11.51718753078989950274209481506, −10.43611438122496292506163436524, −10.01232077567384980778631767676, −9.006797187747200266699837289168, −7.927486761204415861069895977099, −6.97844076720205466342219026804, −5.81753911557302558888992722345, −3.99815483252643344731290175663, −3.03378910265524996368079420260, −2.20812190611208191413700170322, 1.26472004974172380997976075391, 2.11755195125558018642404859050, 3.81037770190537675909303401193, 5.04997949490911655576173692793, 6.36884725790953206409835179953, 7.80466518348520061840683724410, 8.510886226581951680110225649429, 9.217027540517207154367407344420, 9.852722156444235134637425044764, 11.70688995719994455269126414361

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