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Properties

Label 2-2548-2548.1975-c0-0-0
Degree 22
Conductor 25482548
Sign −0.761+0.648i-0.761 + 0.648i
Analytic cond. 1.271611.27161
Root an. cond. 1.127661.12766
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.400 − 0.193i)17-s + 18-s − 0.445·19-s + (−1.12 − 0.541i)22-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)26-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.400 − 0.193i)17-s + 18-s − 0.445·19-s + (−1.12 − 0.541i)22-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)26-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.761+0.648i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.761+0.648i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.761+0.648i-0.761 + 0.648i
Analytic conductor: 1.271611.27161
Root analytic conductor: 1.127661.12766
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2548(1975,⋅)\chi_{2548} (1975, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :0), −0.761+0.648i)(2,\ 2548,\ (\ :0),\ -0.761 + 0.648i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.22184061470.2218406147
L(12)L(\frac12) ≈\approx 0.22184061470.2218406147
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+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
7 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
13 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
good3 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
5 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
11 1+(0.277−1.21i)T+(−0.900−0.433i)T2 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2}
17 1+(−0.400+0.193i)T+(0.623−0.781i)T2 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2}
19 1+0.445T+T2 1 + 0.445T + T^{2}
23 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
29 1+(1.12−0.541i)T+(0.623−0.781i)T2 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2}
31 1+0.445T+T2 1 + 0.445T + T^{2}
37 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
41 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
43 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
47 1+(0.277−1.21i)T+(−0.900−0.433i)T2 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2}
53 1+(1.80+0.867i)T+(0.623+0.781i)T2 1 + (1.80 + 0.867i)T + (0.623 + 0.781i)T^{2}
59 1+(1.12+1.40i)T+(−0.222+0.974i)T2 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2}
61 1+(1.80−0.867i)T+(0.623−0.781i)T2 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2}
67 1+1.80T+T2 1 + 1.80T + T^{2}
71 1+(−0.400−0.193i)T+(0.623+0.781i)T2 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2}
73 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
79 1−T2 1 - T^{2}
83 1+(−0.400−1.75i)T+(−0.900+0.433i)T2 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2}
89 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
97 1−T2 1 - T^{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

−9.420712872276977990395781650921, −8.954794250233909157658867582212, −7.896241634532518845785044772696, −7.17035030250565119061430040107, −6.46535152100749143087983885134, −5.99981446656014712208061861843, −4.92604876084065933709630126148, −4.17611773901845600939195728131, −3.17939759200446260153946576703, −1.79318805726508741604400361597, 0.15215382553788516007572386100, 1.70657103113020274156470119016, 3.00182115033747969205920532618, 3.34774872814815561739776879039, 4.46756231848982927271263747369, 5.48281603889987292930697715989, 6.03751925282809246673194393041, 7.57491358017076586781980085961, 7.79644565682351171793552349572, 8.807601058038574073776651605736

Graph of the ZZ-function along the critical line