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Properties

Label 2-2e8-32.29-c1-0-2
Degree $2$
Conductor $256$
Sign $0.677 - 0.735i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 + 0.943i)3-s + (0.707 + 1.70i)5-s + (−0.665 + 0.665i)7-s + (2.18 + 2.18i)9-s + (−3.69 + 1.52i)11-s + (1.76 − 4.26i)13-s + 4.55i·15-s − 3.61i·17-s + (−0.194 + 0.470i)19-s + (−2.14 + 0.887i)21-s + (−1.33 − 1.33i)23-s + (1.12 − 1.12i)25-s + (0.0793 + 0.191i)27-s + (5.73 + 2.37i)29-s + 1.17·31-s + ⋯
L(s)  = 1  + (1.31 + 0.544i)3-s + (0.316 + 0.763i)5-s + (−0.251 + 0.251i)7-s + (0.726 + 0.726i)9-s + (−1.11 + 0.461i)11-s + (0.489 − 1.18i)13-s + 1.17i·15-s − 0.877i·17-s + (−0.0446 + 0.107i)19-s + (−0.467 + 0.193i)21-s + (−0.278 − 0.278i)23-s + (0.224 − 0.224i)25-s + (0.0152 + 0.0368i)27-s + (1.06 + 0.441i)29-s + 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67029 + 0.732344i\)
\(L(\frac12)\) \(\approx\) \(1.67029 + 0.732344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.27 - 0.943i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.665 - 0.665i)T - 7iT^{2} \)
11 \( 1 + (3.69 - 1.52i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.76 + 4.26i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.61iT - 17T^{2} \)
19 \( 1 + (0.194 - 0.470i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.33 + 1.33i)T + 23iT^{2} \)
29 \( 1 + (-5.73 - 2.37i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (0.510 + 1.23i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.66 - 1.66i)T + 41iT^{2} \)
43 \( 1 + (-2.54 + 1.05i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.49iT - 47T^{2} \)
53 \( 1 + (4.59 - 1.90i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.04 + 4.94i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (13.7 + 5.67i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.40 - 1.41i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.66 - 9.66i)T - 71iT^{2} \)
73 \( 1 + (7.55 + 7.55i)T + 73iT^{2} \)
79 \( 1 + 17.2iT - 79T^{2} \)
83 \( 1 + (4.82 - 11.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.43 - 5.43i)T - 89iT^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−12.33237726444144951665095674154, −10.74714163105570336796922381075, −10.21088264341115487366520611448, −9.310875917545369181562622993160, −8.278745240791381085133875216166, −7.49258656547313816141171353779, −6.10951489616499137833955215784, −4.72997426131601434766826327422, −3.13480849823815688941258066433, −2.61072250907534595217220327359, 1.65217223939414474868361680920, 3.01235023400757348674761997961, 4.37479162408301729588453541661, 5.88121184346834481717863397976, 7.14155904122763631926890493732, 8.248646199624022227795482123459, 8.743822400704179209512716267062, 9.721810163539327057362832596716, 10.85241106493075917471854145437, 12.18903404768883381025804725230

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