↑

Properties

Label 2-2548-91.16-c1-0-43
Degree 22
Conductor 25482548
Sign −0.927+0.374i-0.927 + 0.374i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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)3-s + (1.5 − 2.59i)5-s + (−0.499 − 0.866i)9-s + (−3.5 + 0.866i)13-s + (−3 − 5.19i)15-s + 3·17-s + (−1 − 1.73i)19-s − 6·23-s + (−2 − 3.46i)25-s + 4.00·27-s + (−4.5 − 7.79i)29-s + (−1 − 1.73i)31-s − 7·37-s + (−2 + 6.92i)39-s + (−1.5 − 2.59i)41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.670 − 1.16i)5-s + (−0.166 − 0.288i)9-s + (−0.970 + 0.240i)13-s + (−0.774 − 1.34i)15-s + 0.727·17-s + (−0.229 − 0.397i)19-s − 1.25·23-s + (−0.400 − 0.692i)25-s + 0.769·27-s + (−0.835 − 1.44i)29-s + (−0.179 − 0.311i)31-s − 1.15·37-s + (−0.320 + 1.10i)39-s + (−0.234 − 0.405i)41-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.927+0.374i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((−0.927+0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.927+0.374i-0.927 + 0.374i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(1745,⋅)\chi_{2548} (1745, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), −0.927+0.374i)(2,\ 2548,\ (\ :1/2),\ -0.927 + 0.374i)

Particular Values

L(1)L(1) ≈\approx 1.9594332751.959433275
L(12)L(\frac12) ≈\approx 1.9594332751.959433275
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
13 1+(3.5−0.866i)T 1 + (3.5 - 0.866i)T
good3 1+(−1+1.73i)T+(−1.5−2.59i)T2 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.5+2.59i)T+(−2.5−4.33i)T2 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2}
11 1+(−5.5−9.52i)T2 1 + (-5.5 - 9.52i)T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+(4.5+7.79i)T+(−14.5+25.1i)T2 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+(1+1.73i)T+(−15.5+26.8i)T2 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1+7T+37T2 1 + 7T + 37T^{2}
41 1+(1.5+2.59i)T+(−20.5+35.5i)T2 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2+3.46i)T+(−21.5−37.2i)T2 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3+5.19i)T+(−23.5−40.7i)T2 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.5+7.79i)T+(−26.5+45.8i)T2 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2}
59 1+59T2 1 + 59T^{2}
61 1+(2.5+4.33i)T+(−30.5+52.8i)T2 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(1−1.73i)T+(−33.5−58.0i)T2 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3+5.19i)T+(−35.5−61.4i)T2 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.5−0.866i)T+(−36.5+63.2i)T2 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2+3.46i)T+(−39.5−68.4i)T2 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+(7−12.1i)T+(−48.5−84.0i)T2 1 + (7 - 12.1i)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

−8.452664844453679239189769093960, −7.85696181145068780415712598843, −7.21118423529428943881744979069, −6.28158311034488783967551418528, −5.43536537940392242209814991570, −4.74056786212317888687980613169, −3.64901296651492747226524531439, −2.21151771073269309167050483215, −1.88210027625719964194518484341, −0.54985498454079835692019597952, 1.79229660327676767780509767249, 2.87204148640304126630083330264, 3.39934445629807579001729960452, 4.35624408220353777111013392431, 5.32636932908596543647770963037, 6.10141516748893492264726136130, 6.99470481075728621971243068380, 7.68043368172599761131179078276, 8.619464157148427778335960365904, 9.497489877109192395049182101284

Graph of the ZZ-function along the critical line