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Properties

Label 2-3248-3248.307-c0-0-0
Degree 22
Conductor 32483248
Sign 0.979−0.201i0.979 - 0.201i
Analytic cond. 1.620961.62096
Root an. cond. 1.273171.27317
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 9-s − 14-s + 16-s + 18-s + i·25-s + 28-s + 29-s − 32-s − 36-s + 2·43-s + 49-s − i·50-s + ⋯
L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 9-s − 14-s + 16-s + 18-s + i·25-s + 28-s + 29-s − 32-s − 36-s + 2·43-s + 49-s − i·50-s + ⋯

Functional equation

Λ(s)=(3248s/2 ΓC(s) L(s)=((0.979−0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3248s/2 ΓC(s) L(s)=((0.979−0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32483248    =    24⋅7⋅292^{4} \cdot 7 \cdot 29
Sign: 0.979−0.201i0.979 - 0.201i
Analytic conductor: 1.620961.62096
Root analytic conductor: 1.273171.27317
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3248(307,⋅)\chi_{3248} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3248, ( :0), 0.979−0.201i)(2,\ 3248,\ (\ :0),\ 0.979 - 0.201i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.84668142720.8466814272
L(12)L(\frac12) ≈\approx 0.84668142720.8466814272
L(1)L(1) 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+T 1 + T
7 1−T 1 - T
29 1−T 1 - T
good3 1+T2 1 + T^{2}
5 1−iT2 1 - iT^{2}
11 1+T2 1 + T^{2}
13 1+iT2 1 + iT^{2}
17 1+iT2 1 + iT^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
31 1−iT2 1 - iT^{2}
37 1+T2 1 + T^{2}
41 1+iT2 1 + iT^{2}
43 1−2T+T2 1 - 2T + T^{2}
47 1+iT2 1 + iT^{2}
53 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
59 1−iT2 1 - iT^{2}
61 1−T2 1 - T^{2}
67 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
71 1−2iT−T2 1 - 2iT - T^{2}
73 1+iT2 1 + iT^{2}
79 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
83 1+iT2 1 + iT^{2}
89 1−iT2 1 - iT^{2}
97 1−iT2 1 - iT^{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

−8.669169123759502447910530288252, −8.329973364466046051478228163938, −7.51279116184036266999989923339, −6.86372651829014703807993398501, −5.81616868536283855247807630184, −5.33027175068267797906901932640, −4.13048132632576997271949163170, −2.97326892554484005665052214901, −2.18400660027071355422522475813, −1.02967192569932525265088565312, 0.905411723927944538741942797484, 2.17998584505851809215630760129, 2.85540626242754783418027731214, 4.10313466885390304629178645617, 5.15427182921512863451737905834, 5.93898304515711392371357480592, 6.64153991200497845919031044149, 7.66668530326608183975860618988, 8.072657656966993276945340033109, 8.808022153566740644336177429717

Graph of the ZZ-function along the critical line