↑

Properties

Label 2-392-7.2-c1-0-6
Degree 22
Conductor 392392
Sign 0.701+0.712i0.701 + 0.712i
Analytic cond. 3.130133.13013
Root an. cond. 1.769211.76921
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  + (1 − 1.73i)5-s + (1.5 − 2.59i)9-s + (2 + 3.46i)11-s − 2·13-s + (−3 − 5.19i)17-s + (4 − 6.92i)19-s + (0.500 + 0.866i)25-s + 6·29-s + (4 + 6.92i)31-s + (1 − 1.73i)37-s − 2·41-s − 4·43-s + (−3 − 5.19i)45-s + (−4 + 6.92i)47-s + (−3 − 5.19i)53-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.5 − 0.866i)9-s + (0.603 + 1.04i)11-s − 0.554·13-s + (−0.727 − 1.26i)17-s + (0.917 − 1.58i)19-s + (0.100 + 0.173i)25-s + 1.11·29-s + (0.718 + 1.24i)31-s + (0.164 − 0.284i)37-s − 0.312·41-s − 0.609·43-s + (−0.447 − 0.774i)45-s + (−0.583 + 1.01i)47-s + (−0.412 − 0.713i)53-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((0.701+0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+1/2) L(s)=((0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: 0.701+0.712i0.701 + 0.712i
Analytic conductor: 3.130133.13013
Root analytic conductor: 1.769211.76921
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ392(177,⋅)\chi_{392} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 392, ( :1/2), 0.701+0.712i)(2,\ 392,\ (\ :1/2),\ 0.701 + 0.712i)

Particular Values

L(1)L(1) ≈\approx 1.38639−0.581016i1.38639 - 0.581016i
L(12)L(\frac12) ≈\approx 1.38639−0.581016i1.38639 - 0.581016i
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
7 1 1
good3 1+(−1.5+2.59i)T2 1 + (-1.5 + 2.59i)T^{2}
5 1+(−1+1.73i)T+(−2.5−4.33i)T2 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2−3.46i)T+(−5.5+9.52i)T2 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
17 1+(3+5.19i)T+(−8.5+14.7i)T2 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2}
19 1+(−4+6.92i)T+(−9.5−16.4i)T2 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(−4−6.92i)T+(−15.5+26.8i)T2 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1+1.73i)T+(−18.5−32.0i)T2 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+(4−6.92i)T+(−23.5−40.7i)T2 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2}
53 1+(3+5.19i)T+(−26.5+45.8i)T2 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(3−5.19i)T+(−30.5−52.8i)T2 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2−3.46i)T+(−33.5+58.0i)T2 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(−5−8.66i)T+(−36.5+63.2i)T2 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2}
79 1+(8−13.8i)T+(−39.5−68.4i)T2 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2}
83 1+8T+83T2 1 + 8T + 83T^{2}
89 1+(3−5.19i)T+(−44.5−77.0i)T2 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1−6T+97T2 1 - 6T + 97T^{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

−11.40344511286972910858701886830, −9.917274417527311841030956296240, −9.439663427729552651315334271770, −8.725288688985365401154026504464, −7.14589684481559570253666464729, −6.71110751894114905772863100706, −5.06525128929975497774589015480, −4.52405899190915212718345758041, −2.82041761037243572874998032881, −1.15134921379481204045332394617, 1.80138803823286742340519672064, 3.19527635429439106026589972181, 4.46263616498577605798171719832, 5.86187912253282319375876365814, 6.54780586936613037623635497675, 7.76044616594718079628312705115, 8.539230659751575763907655294321, 9.929124609426543570709263219077, 10.36785676688309124009573448310, 11.32043421468784366133231299063

Graph of the ZZ-function along the critical line