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Properties

Label 2-2e8-32.29-c1-0-2
Degree 22
Conductor 256256
Sign 0.677−0.735i0.677 - 0.735i
Analytic cond. 2.044172.04417
Root an. cond. 1.429741.42974
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(256s/2 ΓC(s) L(s)=((0.677−0.735i) Λ‾(2−s)\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}
Λ(s)=(256s/2 ΓC(s+1/2) L(s)=((0.677−0.735i) Λ‾(1−s)\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: 22
Conductor: 256256    =    282^{8}
Sign: 0.677−0.735i0.677 - 0.735i
Analytic conductor: 2.044172.04417
Root analytic conductor: 1.429741.42974
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ256(161,⋅)\chi_{256} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 256, ( :1/2), 0.677−0.735i)(2,\ 256,\ (\ :1/2),\ 0.677 - 0.735i)

Particular Values

L(1)L(1) ≈\approx 1.67029+0.732344i1.67029 + 0.732344i
L(12)L(\frac12) ≈\approx 1.67029+0.732344i1.67029 + 0.732344i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
good3 1+(−2.27−0.943i)T+(2.12+2.12i)T2 1 + (-2.27 - 0.943i)T + (2.12 + 2.12i)T^{2}
5 1+(−0.707−1.70i)T+(−3.53+3.53i)T2 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2}
7 1+(0.665−0.665i)T−7iT2 1 + (0.665 - 0.665i)T - 7iT^{2}
11 1+(3.69−1.52i)T+(7.77−7.77i)T2 1 + (3.69 - 1.52i)T + (7.77 - 7.77i)T^{2}
13 1+(−1.76+4.26i)T+(−9.19−9.19i)T2 1 + (-1.76 + 4.26i)T + (-9.19 - 9.19i)T^{2}
17 1+3.61iT−17T2 1 + 3.61iT - 17T^{2}
19 1+(0.194−0.470i)T+(−13.4−13.4i)T2 1 + (0.194 - 0.470i)T + (-13.4 - 13.4i)T^{2}
23 1+(1.33+1.33i)T+23iT2 1 + (1.33 + 1.33i)T + 23iT^{2}
29 1+(−5.73−2.37i)T+(20.5+20.5i)T2 1 + (-5.73 - 2.37i)T + (20.5 + 20.5i)T^{2}
31 1−1.17T+31T2 1 - 1.17T + 31T^{2}
37 1+(0.510+1.23i)T+(−26.1+26.1i)T2 1 + (0.510 + 1.23i)T + (-26.1 + 26.1i)T^{2}
41 1+(−1.66−1.66i)T+41iT2 1 + (-1.66 - 1.66i)T + 41iT^{2}
43 1+(−2.54+1.05i)T+(30.4−30.4i)T2 1 + (-2.54 + 1.05i)T + (30.4 - 30.4i)T^{2}
47 1−1.49iT−47T2 1 - 1.49iT - 47T^{2}
53 1+(4.59−1.90i)T+(37.4−37.4i)T2 1 + (4.59 - 1.90i)T + (37.4 - 37.4i)T^{2}
59 1+(2.04+4.94i)T+(−41.7+41.7i)T2 1 + (2.04 + 4.94i)T + (-41.7 + 41.7i)T^{2}
61 1+(13.7+5.67i)T+(43.1+43.1i)T2 1 + (13.7 + 5.67i)T + (43.1 + 43.1i)T^{2}
67 1+(−3.40−1.41i)T+(47.3+47.3i)T2 1 + (-3.40 - 1.41i)T + (47.3 + 47.3i)T^{2}
71 1+(9.66−9.66i)T−71iT2 1 + (9.66 - 9.66i)T - 71iT^{2}
73 1+(7.55+7.55i)T+73iT2 1 + (7.55 + 7.55i)T + 73iT^{2}
79 1+17.2iT−79T2 1 + 17.2iT - 79T^{2}
83 1+(4.82−11.6i)T+(−58.6−58.6i)T2 1 + (4.82 - 11.6i)T + (-58.6 - 58.6i)T^{2}
89 1+(5.43−5.43i)T−89iT2 1 + (5.43 - 5.43i)T - 89iT^{2}
97 1−6.15T+97T2 1 - 6.15T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line