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Properties

Label 2-2e5-4.3-c8-0-7
Degree 22
Conductor 3232
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 13.036113.0361
Root an. cond. 3.610553.61055
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 135. i·3-s − 581.·5-s − 2.67e3i·7-s − 1.19e4·9-s + 1.41e4i·11-s + 4.28e4·13-s + 7.90e4i·15-s − 1.40e5·17-s + 1.29e5i·19-s − 3.62e5·21-s − 1.96e5i·23-s − 5.22e4·25-s + 7.27e5i·27-s − 3.55e5·29-s + 3.35e3i·31-s + ⋯
L(s)  = 1  − 1.67i·3-s − 0.930·5-s − 1.11i·7-s − 1.81·9-s + 0.966i·11-s + 1.49·13-s + 1.56i·15-s − 1.68·17-s + 0.995i·19-s − 1.86·21-s − 0.703i·23-s − 0.133·25-s + 1.36i·27-s − 0.503·29-s + 0.00363i·31-s + ⋯

Functional equation

Λ(s)=(32s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(32s/2 ΓC(s+4) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3232    =    252^{5}
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 13.036113.0361
Root analytic conductor: 3.610553.61055
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ32(31,⋅)\chi_{32} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 32, ( :4), −0.707−0.707i)(2,\ 32,\ (\ :4),\ -0.707 - 0.707i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.229040+0.552952i0.229040 + 0.552952i
L(12)L(\frac12) ≈\approx 0.229040+0.552952i0.229040 + 0.552952i
L(5)L(5) 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+135.iT−6.56e3T2 1 + 135. iT - 6.56e3T^{2}
5 1+581.T+3.90e5T2 1 + 581.T + 3.90e5T^{2}
7 1+2.67e3iT−5.76e6T2 1 + 2.67e3iT - 5.76e6T^{2}
11 1−1.41e4iT−2.14e8T2 1 - 1.41e4iT - 2.14e8T^{2}
13 1−4.28e4T+8.15e8T2 1 - 4.28e4T + 8.15e8T^{2}
17 1+1.40e5T+6.97e9T2 1 + 1.40e5T + 6.97e9T^{2}
19 1−1.29e5iT−1.69e10T2 1 - 1.29e5iT - 1.69e10T^{2}
23 1+1.96e5iT−7.83e10T2 1 + 1.96e5iT - 7.83e10T^{2}
29 1+3.55e5T+5.00e11T2 1 + 3.55e5T + 5.00e11T^{2}
31 1−3.35e3iT−8.52e11T2 1 - 3.35e3iT - 8.52e11T^{2}
37 1−9.07e5T+3.51e12T2 1 - 9.07e5T + 3.51e12T^{2}
41 1+3.06e6T+7.98e12T2 1 + 3.06e6T + 7.98e12T^{2}
43 1+5.01e6iT−1.16e13T2 1 + 5.01e6iT - 1.16e13T^{2}
47 1+3.25e6iT−2.38e13T2 1 + 3.25e6iT - 2.38e13T^{2}
53 1−2.34e5T+6.22e13T2 1 - 2.34e5T + 6.22e13T^{2}
59 1+7.78e6iT−1.46e14T2 1 + 7.78e6iT - 1.46e14T^{2}
61 1+2.42e7T+1.91e14T2 1 + 2.42e7T + 1.91e14T^{2}
67 1+6.87e6iT−4.06e14T2 1 + 6.87e6iT - 4.06e14T^{2}
71 1+5.76e6iT−6.45e14T2 1 + 5.76e6iT - 6.45e14T^{2}
73 1+1.19e7T+8.06e14T2 1 + 1.19e7T + 8.06e14T^{2}
79 1−3.55e7iT−1.51e15T2 1 - 3.55e7iT - 1.51e15T^{2}
83 1+2.03e7iT−2.25e15T2 1 + 2.03e7iT - 2.25e15T^{2}
89 1+1.19e7T+3.93e15T2 1 + 1.19e7T + 3.93e15T^{2}
97 1+3.19e7T+7.83e15T2 1 + 3.19e7T + 7.83e15T^{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

−13.85187390250110299469149861654, −13.06854448663756853344112324211, −11.93824628558070040382219895465, −10.79311460963032039208021045444, −8.462325418541455206241791374352, −7.41715028560302545998238239567, −6.49194101977894172402773192351, −3.98413711600608749817441039965, −1.72129451089016326807996198709, −0.25328177459807282278534473988, 3.18369814818584980981364185652, 4.41460824133692604749422474687, 5.93537970653782596092755562246, 8.502570113137408173272631281172, 9.180300247733256859018160395106, 11.04686666148097301715199130532, 11.42694258607022845690115440073, 13.48297849847818767015006704964, 15.24912527072710623246610824620, 15.57206695491111864912712524943

Graph of the ZZ-function along the critical line