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Properties

Label 2-880-880.549-c0-0-3
Degree 22
Conductor 880880
Sign 0.923+0.382i0.923 + 0.382i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.707 − 0.707i)5-s − 1.84i·7-s + (−0.923 − 0.382i)8-s − i·9-s + (0.382 − 0.923i)10-s + (0.707 + 0.707i)11-s + (0.541 − 0.541i)13-s + (1.70 − 0.707i)14-s − i·16-s − 1.84·17-s + (0.923 − 0.382i)18-s + 20-s + (−0.382 + 0.923i)22-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.707 − 0.707i)5-s − 1.84i·7-s + (−0.923 − 0.382i)8-s − i·9-s + (0.382 − 0.923i)10-s + (0.707 + 0.707i)11-s + (0.541 − 0.541i)13-s + (1.70 − 0.707i)14-s − i·16-s − 1.84·17-s + (0.923 − 0.382i)18-s + 20-s + (−0.382 + 0.923i)22-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 0.923+0.382i0.923 + 0.382i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ880(549,⋅)\chi_{880} (549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 880, ( :0), 0.923+0.382i)(2,\ 880,\ (\ :0),\ 0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91152438620.9115243862
L(12)L(\frac12) ≈\approx 0.91152438620.9115243862
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+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
11 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good3 1+iT2 1 + iT^{2}
7 1+1.84iT−T2 1 + 1.84iT - T^{2}
13 1+(−0.541+0.541i)T−iT2 1 + (-0.541 + 0.541i)T - iT^{2}
17 1+1.84T+T2 1 + 1.84T + T^{2}
19 1+iT2 1 + iT^{2}
23 1+T2 1 + T^{2}
29 1+iT2 1 + iT^{2}
31 1−1.41T+T2 1 - 1.41T + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1+(−0.541−0.541i)T+iT2 1 + (-0.541 - 0.541i)T + iT^{2}
47 1−T2 1 - T^{2}
53 1−iT2 1 - iT^{2}
59 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
61 1+iT2 1 + iT^{2}
67 1+iT2 1 + iT^{2}
71 1−T2 1 - T^{2}
73 1+0.765iT−T2 1 + 0.765iT - T^{2}
79 1−T2 1 - T^{2}
83 1+(−0.541+0.541i)T−iT2 1 + (-0.541 + 0.541i)T - iT^{2}
89 1+1.41iT−T2 1 + 1.41iT - 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

−10.14116467977209160856810274355, −9.194834425369681202046987078405, −8.496079846234672744608736262822, −7.55520798004073183439004841688, −6.88806292864585451565597393224, −6.23238337400201302253926947275, −4.61028788055395280777988256512, −4.26095014235518550331410840573, −3.47199629017169076450937678846, −0.849011788600262656333846009145, 2.09254557958524630802819860473, 2.77279684190790592427889792415, 3.97619091097436231611158366179, 4.91201159245240035122019326362, 5.99956897193382390726023431235, 6.66861297413990225866766768659, 8.381361929225828828507498587966, 8.652363280369511581114386973629, 9.589675390935455025418666690046, 10.78454434378518778108125486067

Graph of the ZZ-function along the critical line