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Properties

Label 2-3248-3248.1259-c0-0-0
Degree 22
Conductor 32483248
Sign 0.201−0.979i0.201 - 0.979i
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  + i·2-s − 4-s − 7-s − i·8-s − 9-s + 2·11-s − i·14-s + 16-s − i·18-s + 2i·22-s − i·25-s + 28-s + i·29-s + i·32-s + 36-s + 2·37-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 7-s − i·8-s − 9-s + 2·11-s − i·14-s + 16-s − i·18-s + 2i·22-s − i·25-s + 28-s + i·29-s + i·32-s + 36-s + 2·37-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 32483248    =    24⋅7⋅292^{4} \cdot 7 \cdot 29
Sign: 0.201−0.979i0.201 - 0.979i
Analytic conductor: 1.620961.62096
Root analytic conductor: 1.273171.27317
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3248(1259,⋅)\chi_{3248} (1259, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3248, ( :0), 0.201−0.979i)(2,\ 3248,\ (\ :0),\ 0.201 - 0.979i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.99961400160.9996140016
L(12)L(\frac12) ≈\approx 0.99961400160.9996140016
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−iT 1 - iT
7 1+T 1 + T
29 1−iT 1 - iT
good3 1+T2 1 + T^{2}
5 1+iT2 1 + iT^{2}
11 1−2T+T2 1 - 2T + 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 1−2T+T2 1 - 2T + T^{2}
41 1−iT2 1 - iT^{2}
43 1+T2 1 + 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

−8.917921474910340880417822593487, −8.359491086471494577215781495456, −7.34061356570875694463317342982, −6.56059215208243323846002820976, −6.22100396553270048216950933609, −5.47067254707000836751743148612, −4.30516417242503364543832677690, −3.73475825308054763714285040756, −2.74455512367905841630091617610, −0.965233113394577134863110134092, 0.848475870737398988176698540593, 2.13171602887561559556259490452, 3.14137697108325333499900137927, 3.75058086078739729740128362060, 4.51269149323768553180447434665, 5.74654412253157256342506509526, 6.19987342142849872194238143306, 7.15506385365185897838200250574, 8.236280667627223465900401460496, 8.959911730797695258485378241839

Graph of the ZZ-function along the critical line