↑

Properties

Label 2-72-72.13-c1-0-5
Degree 22
Conductor 7272
Sign 0.657−0.753i0.657 - 0.753i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
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  + (0.587 + 1.28i)2-s + (1.69 − 0.378i)3-s + (−1.30 + 1.51i)4-s + (−1.97 − 1.14i)5-s + (1.47 + 1.95i)6-s + (−0.907 − 1.57i)7-s + (−2.71 − 0.795i)8-s + (2.71 − 1.27i)9-s + (0.306 − 3.21i)10-s + (−4.24 + 2.44i)11-s + (−1.64 + 3.05i)12-s + (4.00 + 2.31i)13-s + (1.48 − 2.09i)14-s + (−3.77 − 1.18i)15-s + (−0.570 − 3.95i)16-s + 1.92·17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.975 − 0.218i)3-s + (−0.654 + 0.755i)4-s + (−0.883 − 0.510i)5-s + (0.604 + 0.796i)6-s + (−0.343 − 0.594i)7-s + (−0.959 − 0.281i)8-s + (0.904 − 0.426i)9-s + (0.0968 − 1.01i)10-s + (−1.27 + 0.738i)11-s + (−0.473 + 0.880i)12-s + (1.11 + 0.641i)13-s + (0.398 − 0.559i)14-s + (−0.973 − 0.304i)15-s + (−0.142 − 0.989i)16-s + 0.467·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.657−0.753i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.657−0.753i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.657−0.753i0.657 - 0.753i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(13,⋅)\chi_{72} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), 0.657−0.753i)(2,\ 72,\ (\ :1/2),\ 0.657 - 0.753i)

Particular Values

L(1)L(1) ≈\approx 1.07287+0.487486i1.07287 + 0.487486i
L(12)L(\frac12) ≈\approx 1.07287+0.487486i1.07287 + 0.487486i
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+(−0.587−1.28i)T 1 + (-0.587 - 1.28i)T
3 1+(−1.69+0.378i)T 1 + (-1.69 + 0.378i)T
good5 1+(1.97+1.14i)T+(2.5+4.33i)T2 1 + (1.97 + 1.14i)T + (2.5 + 4.33i)T^{2}
7 1+(0.907+1.57i)T+(−3.5+6.06i)T2 1 + (0.907 + 1.57i)T + (-3.5 + 6.06i)T^{2}
11 1+(4.24−2.44i)T+(5.5−9.52i)T2 1 + (4.24 - 2.44i)T + (5.5 - 9.52i)T^{2}
13 1+(−4.00−2.31i)T+(6.5+11.2i)T2 1 + (-4.00 - 2.31i)T + (6.5 + 11.2i)T^{2}
17 1−1.92T+17T2 1 - 1.92T + 17T^{2}
19 1+2.12iT−19T2 1 + 2.12iT - 19T^{2}
23 1+(1.15−2.00i)T+(−11.5−19.9i)T2 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.16+1.82i)T+(14.5−25.1i)T2 1 + (-3.16 + 1.82i)T + (14.5 - 25.1i)T^{2}
31 1+(2.65−4.60i)T+(−15.5−26.8i)T2 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2}
37 1−7.98iT−37T2 1 - 7.98iT - 37T^{2}
41 1+(2.36−4.09i)T+(−20.5−35.5i)T2 1 + (2.36 - 4.09i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.20−1.27i)T+(21.5−37.2i)T2 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2}
47 1+(2.02+3.49i)T+(−23.5+40.7i)T2 1 + (2.02 + 3.49i)T + (-23.5 + 40.7i)T^{2}
53 1+8.95iT−53T2 1 + 8.95iT - 53T^{2}
59 1+(−3.05−1.76i)T+(29.5+51.0i)T2 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2}
61 1+(−1.71+0.991i)T+(30.5−52.8i)T2 1 + (-1.71 + 0.991i)T + (30.5 - 52.8i)T^{2}
67 1+(7.72+4.46i)T+(33.5+58.0i)T2 1 + (7.72 + 4.46i)T + (33.5 + 58.0i)T^{2}
71 1−13.3T+71T2 1 - 13.3T + 71T^{2}
73 1+11.5T+73T2 1 + 11.5T + 73T^{2}
79 1+(4.97+8.61i)T+(−39.5+68.4i)T2 1 + (4.97 + 8.61i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.12+1.80i)T+(41.5−71.8i)T2 1 + (-3.12 + 1.80i)T + (41.5 - 71.8i)T^{2}
89 1−2.49T+89T2 1 - 2.49T + 89T^{2}
97 1+(−6.99−12.1i)T+(−48.5+84.0i)T2 1 + (-6.99 - 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

−14.91802377515106952169637665344, −13.62865613062889630224978825829, −13.09422987969527482029447139767, −11.95254456046828480057570097692, −9.982080615838929807857780270381, −8.582089976265332008898049886723, −7.82605066229181761490246488452, −6.77717620251028244551546295518, −4.70070231611120625295778626878, −3.48748889216191682597875390263, 2.82368472877987340325548906309, 3.78299947695915591829500263405, 5.68312454052873255757110201877, 7.85702128582173897228789963487, 8.824375187601826531347937934966, 10.27357586118884874064143799284, 11.05199725887102069148058036778, 12.46950248130481375709006721605, 13.34373313837765158441049972729, 14.38617051101068374344458415225

Graph of the ZZ-function along the critical line