↑

Properties

Label 2-21e2-63.25-c1-0-19
Degree 22
Conductor 441441
Sign −0.944+0.327i-0.944 + 0.327i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  − 2.71·2-s + (−1.16 − 1.27i)3-s + 5.37·4-s + (−0.793 + 1.37i)5-s + (3.17 + 3.46i)6-s − 9.15·8-s + (−0.264 + 2.98i)9-s + (2.15 − 3.73i)10-s + (0.674 + 1.16i)11-s + (−6.28 − 6.86i)12-s + (−1.58 − 2.75i)13-s + (2.68 − 0.593i)15-s + 14.1·16-s + (1.40 − 2.42i)17-s + (0.717 − 8.11i)18-s + (0.312 + 0.541i)19-s + ⋯
L(s)  = 1  − 1.91·2-s + (−0.675 − 0.737i)3-s + 2.68·4-s + (−0.354 + 0.614i)5-s + (1.29 + 1.41i)6-s − 3.23·8-s + (−0.0880 + 0.996i)9-s + (0.681 − 1.17i)10-s + (0.203 + 0.352i)11-s + (−1.81 − 1.98i)12-s + (−0.440 − 0.763i)13-s + (0.692 − 0.153i)15-s + 3.52·16-s + (0.339 − 0.588i)17-s + (0.169 − 1.91i)18-s + (0.0717 + 0.124i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.944+0.327i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((−0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.944+0.327i-0.944 + 0.327i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(214,⋅)\chi_{441} (214, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), −0.944+0.327i)(2,\ 441,\ (\ :1/2),\ -0.944 + 0.327i)

Particular Values

L(1)L(1) ≈\approx 0.0206598−0.122739i0.0206598 - 0.122739i
L(12)L(\frac12) ≈\approx 0.0206598−0.122739i0.0206598 - 0.122739i
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)
bad3 1+(1.16+1.27i)T 1 + (1.16 + 1.27i)T
7 1 1
good2 1+2.71T+2T2 1 + 2.71T + 2T^{2}
5 1+(0.793−1.37i)T+(−2.5−4.33i)T2 1 + (0.793 - 1.37i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.674−1.16i)T+(−5.5+9.52i)T2 1 + (-0.674 - 1.16i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.58+2.75i)T+(−6.5+11.2i)T2 1 + (1.58 + 2.75i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1.40+2.42i)T+(−8.5−14.7i)T2 1 + (-1.40 + 2.42i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.312−0.541i)T+(−9.5+16.4i)T2 1 + (-0.312 - 0.541i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.142+0.246i)T+(−11.5−19.9i)T2 1 + (-0.142 + 0.246i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.27+3.93i)T+(−14.5−25.1i)T2 1 + (-2.27 + 3.93i)T + (-14.5 - 25.1i)T^{2}
31 1+7.43T+31T2 1 + 7.43T + 31T^{2}
37 1+(4.01+6.94i)T+(−18.5+32.0i)T2 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2}
41 1+(5.01+8.68i)T+(−20.5+35.5i)T2 1 + (5.01 + 8.68i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.12−5.42i)T+(−21.5−37.2i)T2 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2}
47 1+11.1T+47T2 1 + 11.1T + 47T^{2}
53 1+(1.39−2.41i)T+(−26.5−45.8i)T2 1 + (1.39 - 2.41i)T + (-26.5 - 45.8i)T^{2}
59 1+4.57T+59T2 1 + 4.57T + 59T^{2}
61 1−0.385T+61T2 1 - 0.385T + 61T^{2}
67 1+2.53T+67T2 1 + 2.53T + 67T^{2}
71 1+1.45T+71T2 1 + 1.45T + 71T^{2}
73 1+(0.234−0.405i)T+(−36.5−63.2i)T2 1 + (0.234 - 0.405i)T + (-36.5 - 63.2i)T^{2}
79 1+15.7T+79T2 1 + 15.7T + 79T^{2}
83 1+(−6.99+12.1i)T+(−41.5−71.8i)T2 1 + (-6.99 + 12.1i)T + (-41.5 - 71.8i)T^{2}
89 1+(1.29+2.24i)T+(−44.5+77.0i)T2 1 + (1.29 + 2.24i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.22−12.5i)T+(−48.5−84.0i)T2 1 + (7.22 - 12.5i)T + (-48.5 - 84.0i)T^{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

−10.61681274283055071188693325557, −9.936927129410327810169976795345, −8.867760415760188118386575030581, −7.75605673252296508388075475765, −7.34093886465864239640052753175, −6.56042036544171933119319256002, −5.43815568568833874331187228045, −3.03197426835612038589858940981, −1.75028749065787570011280460888, −0.16284889465615840253086987846, 1.43162869545379882977072271589, 3.35292805188748226509633989840, 4.94036915453299441992987875694, 6.25796755457878818196564039801, 7.03796627074153640747916038236, 8.299447030611707471315889221533, 8.861746629986732417420681961995, 9.733288851872664478742081206699, 10.37118736000755067139151583659, 11.31018017509795964472516880664

Graph of the ZZ-function along the critical line