↑

Properties

Label 2-416-104.99-c1-0-0
Degree 22
Conductor 416416
Sign −0.957−0.289i-0.957 - 0.289i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−2.54 + 2.54i)5-s + (−2.54 − 2.54i)7-s − 2·9-s + (−1 + i)11-s + (−2.54 + 2.54i)13-s + (−2.54 + 2.54i)15-s + 3i·17-s + (−2 − 2i)19-s + (−2.54 − 2.54i)21-s + 5.09·23-s − 7.99i·25-s − 5·27-s − 5.09i·29-s + (−5.09 + 5.09i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−1.14 + 1.14i)5-s + (−0.963 − 0.963i)7-s − 0.666·9-s + (−0.301 + 0.301i)11-s + (−0.707 + 0.707i)13-s + (−0.658 + 0.658i)15-s + 0.727i·17-s + (−0.458 − 0.458i)19-s + (−0.556 − 0.556i)21-s + 1.06·23-s − 1.59i·25-s − 0.962·27-s − 0.946i·29-s + (−0.915 + 0.915i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.957−0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.957−0.289i-0.957 - 0.289i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(47,⋅)\chi_{416} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.957−0.289i)(2,\ 416,\ (\ :1/2),\ -0.957 - 0.289i)

Particular Values

L(1)L(1) ≈\approx 0.0516898+0.349093i0.0516898 + 0.349093i
L(12)L(\frac12) ≈\approx 0.0516898+0.349093i0.0516898 + 0.349093i
L(32)L(\frac{3}{2}) 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
13 1+(2.54−2.54i)T 1 + (2.54 - 2.54i)T
good3 1−T+3T2 1 - T + 3T^{2}
5 1+(2.54−2.54i)T−5iT2 1 + (2.54 - 2.54i)T - 5iT^{2}
7 1+(2.54+2.54i)T+7iT2 1 + (2.54 + 2.54i)T + 7iT^{2}
11 1+(1−i)T−11iT2 1 + (1 - i)T - 11iT^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1+(2+2i)T+19iT2 1 + (2 + 2i)T + 19iT^{2}
23 1−5.09T+23T2 1 - 5.09T + 23T^{2}
29 1+5.09iT−29T2 1 + 5.09iT - 29T^{2}
31 1+(5.09−5.09i)T−31iT2 1 + (5.09 - 5.09i)T - 31iT^{2}
37 1+(−2.54−2.54i)T+37iT2 1 + (-2.54 - 2.54i)T + 37iT^{2}
41 1+(−6−6i)T+41iT2 1 + (-6 - 6i)T + 41iT^{2}
43 1+iT−43T2 1 + iT - 43T^{2}
47 1+(2.54+2.54i)T+47iT2 1 + (2.54 + 2.54i)T + 47iT^{2}
53 1−5.09iT−53T2 1 - 5.09iT - 53T^{2}
59 1+(8−8i)T−59iT2 1 + (8 - 8i)T - 59iT^{2}
61 1−61T2 1 - 61T^{2}
67 1+(3+3i)T+67iT2 1 + (3 + 3i)T + 67iT^{2}
71 1+(−7.64+7.64i)T−71iT2 1 + (-7.64 + 7.64i)T - 71iT^{2}
73 1+(6−6i)T−73iT2 1 + (6 - 6i)T - 73iT^{2}
79 1−5.09iT−79T2 1 - 5.09iT - 79T^{2}
83 1+(5+5i)T+83iT2 1 + (5 + 5i)T + 83iT^{2}
89 1+(2−2i)T−89iT2 1 + (2 - 2i)T - 89iT^{2}
97 1+(7+7i)T+97iT2 1 + (7 + 7i)T + 97iT^{2}
show more
show less
   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

−11.40898402206881439816295049502, −10.78775943778824226801881685413, −9.894739706812234225542328830008, −8.865841785925640623721378297013, −7.71442207920456603963892291435, −7.14328859976291820371974682354, −6.29858617194998800922041737282, −4.44603574149262605827057381643, −3.48804977272946765309609836342, −2.67146770059946876187202292332, 0.19767752283746219961372471541, 2.65575089887556229422964292453, 3.57496313961042972605135272492, 4.99132920166081112136667617576, 5.81671496112109025206349593107, 7.37186392693435805117818798283, 8.183503863471162482817469421664, 8.976761544711955104843657128909, 9.465172401269568065799750120225, 10.96625780926382195448254554280

Graph of the ZZ-function along the critical line