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Properties

Label 2-2e8-32.19-c2-0-9
Degree $2$
Conductor $256$
Sign $0.965 - 0.260i$
Analytic cond. $6.97549$
Root an. cond. $2.64111$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.68 + 1.94i)3-s + (4.51 − 1.86i)5-s + (−3.85 − 3.85i)7-s + (11.8 + 11.8i)9-s + (4.56 − 1.89i)11-s + (5.58 + 2.31i)13-s + 24.7·15-s − 25.0i·17-s + (−6.43 + 15.5i)19-s + (−10.5 − 25.5i)21-s + (−26.9 + 26.9i)23-s + (−0.825 + 0.825i)25-s + (15.0 + 36.2i)27-s + (0.210 − 0.507i)29-s + 15.8i·31-s + ⋯
L(s)  = 1  + (1.56 + 0.647i)3-s + (0.902 − 0.373i)5-s + (−0.550 − 0.550i)7-s + (1.31 + 1.31i)9-s + (0.414 − 0.171i)11-s + (0.429 + 0.177i)13-s + 1.65·15-s − 1.47i·17-s + (−0.338 + 0.817i)19-s + (−0.503 − 1.21i)21-s + (−1.16 + 1.16i)23-s + (−0.0330 + 0.0330i)25-s + (0.556 + 1.34i)27-s + (0.00724 − 0.0174i)29-s + 0.510i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.965 - 0.260i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ 0.965 - 0.260i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.83874 + 0.375996i\)
\(L(\frac12)\) \(\approx\) \(2.83874 + 0.375996i\)
\(L(2)\) 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 \( 1 + (-4.68 - 1.94i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-4.51 + 1.86i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (3.85 + 3.85i)T + 49iT^{2} \)
11 \( 1 + (-4.56 + 1.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-5.58 - 2.31i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 25.0iT - 289T^{2} \)
19 \( 1 + (6.43 - 15.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (26.9 - 26.9i)T - 529iT^{2} \)
29 \( 1 + (-0.210 + 0.507i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 15.8iT - 961T^{2} \)
37 \( 1 + (-2.18 + 0.905i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (31.1 + 31.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (12.9 - 5.34i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 15.0T + 2.20e3T^{2} \)
53 \( 1 + (-15.4 - 37.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-14.7 - 35.5i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-15.4 + 37.3i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (61.3 + 25.4i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (51.7 + 51.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-64.9 - 64.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 38.1T + 6.24e3T^{2} \)
83 \( 1 + (15.9 - 38.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-23.7 + 23.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 118.T + 9.40e3T^{2} \)
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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

−11.89512723431982859878899664505, −10.41079665399561036324022209470, −9.664324239067537040795667479949, −9.175233728065249759183637046917, −8.165639209466548761099412437827, −7.04352031647817293542939054584, −5.61515087868466756551824395950, −4.15495858568233695191383245860, −3.22813997383426667869365979947, −1.79717567478630265181561869533, 1.83976767404864381136740968306, 2.71721479168787810804167379497, 3.97917778941289470162220101454, 6.07076203878275012601256706297, 6.69157061242083636182528084178, 8.084573789219896074596552149215, 8.758265408879497929080980175249, 9.648593694319716423942354282223, 10.47987594837369182022040687796, 12.08874771300924835570804926821

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