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Properties

Label 2-2496-312.83-c0-0-0
Degree 22
Conductor 24962496
Sign 0.471−0.881i0.471 - 0.881i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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  − 3-s + (−1 + i)7-s + 9-s − i·13-s + (1 − i)19-s + (1 − i)21-s + i·25-s − 27-s + (1 + i)31-s + (−1 + i)37-s + i·39-s + 2i·43-s − i·49-s + (−1 + i)57-s + 2i·61-s + ⋯
L(s)  = 1  − 3-s + (−1 + i)7-s + 9-s − i·13-s + (1 − i)19-s + (1 − i)21-s + i·25-s − 27-s + (1 + i)31-s + (−1 + i)37-s + i·39-s + 2i·43-s − i·49-s + (−1 + i)57-s + 2i·61-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=((0.471−0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s) L(s)=((0.471−0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 0.471−0.881i0.471 - 0.881i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2496(863,⋅)\chi_{2496} (863, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2496, ( :0), 0.471−0.881i)(2,\ 2496,\ (\ :0),\ 0.471 - 0.881i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69325298880.6932529888
L(12)L(\frac12) ≈\approx 0.69325298880.6932529888
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 1
3 1+T 1 + T
13 1+iT 1 + iT
good5 1−iT2 1 - iT^{2}
7 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
11 1−iT2 1 - iT^{2}
17 1+T2 1 + T^{2}
19 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1+iT2 1 + iT^{2}
43 1−2iT−T2 1 - 2iT - T^{2}
47 1+iT2 1 + iT^{2}
53 1+T2 1 + T^{2}
59 1−iT2 1 - iT^{2}
61 1−2iT−T2 1 - 2iT - T^{2}
67 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
71 1−iT2 1 - iT^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1+iT2 1 + iT^{2}
89 1−iT2 1 - iT^{2}
97 1+(−1+i)T−iT2 1 + (-1 + i)T - 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

−9.406341518427697898461927824375, −8.558885721645276825035866258434, −7.55438943503388609809169084480, −6.75680233350580324166495678319, −6.12702657750630376379904436122, −5.34098408582906122664413196981, −4.83094797818488488928753565992, −3.41348833879036424817153220918, −2.73869407454204586645458863627, −1.13384196265056487221457091608, 0.62333462382742577023659708620, 2.00750035431787600840391285944, 3.58734273369507363531222204989, 4.08998479234489716621711418651, 5.09098650939926648015863496159, 5.99948092411251378989704605809, 6.66345525611340441945464359417, 7.19729776648873898314850890504, 8.036966417482462788692381946811, 9.232710527115376021143863825471

Graph of the ZZ-function along the critical line