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Properties

Label 2-72-72.11-c1-0-1
Degree 22
Conductor 7272
Sign −0.329−0.944i-0.329 - 0.944i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.409 + 1.35i)2-s + (−1.12 + 1.31i)3-s + (−1.66 + 1.10i)4-s + (−0.565 − 0.978i)5-s + (−2.24 − 0.988i)6-s + (3.71 + 2.14i)7-s + (−2.18 − 1.79i)8-s + (−0.456 − 2.96i)9-s + (1.09 − 1.16i)10-s + (1.00 + 0.582i)11-s + (0.419 − 3.43i)12-s + (2.64 − 1.52i)13-s + (−1.38 + 5.90i)14-s + (1.92 + 0.360i)15-s + (1.54 − 3.69i)16-s + 1.49i·17-s + ⋯
L(s)  = 1  + (0.289 + 0.957i)2-s + (−0.651 + 0.758i)3-s + (−0.832 + 0.554i)4-s + (−0.252 − 0.437i)5-s + (−0.915 − 0.403i)6-s + (1.40 + 0.810i)7-s + (−0.771 − 0.636i)8-s + (−0.152 − 0.988i)9-s + (0.345 − 0.368i)10-s + (0.304 + 0.175i)11-s + (0.121 − 0.992i)12-s + (0.733 − 0.423i)13-s + (−0.369 + 1.57i)14-s + (0.496 + 0.0932i)15-s + (0.385 − 0.922i)16-s + 0.362i·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.329−0.944i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((−0.329−0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.329−0.944i-0.329 - 0.944i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(11,⋅)\chi_{72} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), −0.329−0.944i)(2,\ 72,\ (\ :1/2),\ -0.329 - 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.503793+0.709479i0.503793 + 0.709479i
L(12)L(\frac12) ≈\approx 0.503793+0.709479i0.503793 + 0.709479i
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+(−0.409−1.35i)T 1 + (-0.409 - 1.35i)T
3 1+(1.12−1.31i)T 1 + (1.12 - 1.31i)T
good5 1+(0.565+0.978i)T+(−2.5+4.33i)T2 1 + (0.565 + 0.978i)T + (-2.5 + 4.33i)T^{2}
7 1+(−3.71−2.14i)T+(3.5+6.06i)T2 1 + (-3.71 - 2.14i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.00−0.582i)T+(5.5+9.52i)T2 1 + (-1.00 - 0.582i)T + (5.5 + 9.52i)T^{2}
13 1+(−2.64+1.52i)T+(6.5−11.2i)T2 1 + (-2.64 + 1.52i)T + (6.5 - 11.2i)T^{2}
17 1−1.49iT−17T2 1 - 1.49iT - 17T^{2}
19 1+3.42T+19T2 1 + 3.42T + 19T^{2}
23 1+(3.85+6.68i)T+(−11.5+19.9i)T2 1 + (3.85 + 6.68i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.709−1.22i)T+(−14.5−25.1i)T2 1 + (0.709 - 1.22i)T + (-14.5 - 25.1i)T^{2}
31 1+(4.66−2.69i)T+(15.5−26.8i)T2 1 + (4.66 - 2.69i)T + (15.5 - 26.8i)T^{2}
37 1+2.97iT−37T2 1 + 2.97iT - 37T^{2}
41 1+(4.23−2.44i)T+(20.5−35.5i)T2 1 + (4.23 - 2.44i)T + (20.5 - 35.5i)T^{2}
43 1+(1.74−3.01i)T+(−21.5−37.2i)T2 1 + (1.74 - 3.01i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.77−3.08i)T+(−23.5−40.7i)T2 1 + (1.77 - 3.08i)T + (-23.5 - 40.7i)T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1+(−7.50+4.33i)T+(29.5−51.0i)T2 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2}
61 1+(3.16+1.82i)T+(30.5+52.8i)T2 1 + (3.16 + 1.82i)T + (30.5 + 52.8i)T^{2}
67 1+(5.58+9.66i)T+(−33.5+58.0i)T2 1 + (5.58 + 9.66i)T + (-33.5 + 58.0i)T^{2}
71 1−2.54T+71T2 1 - 2.54T + 71T^{2}
73 1+7.06T+73T2 1 + 7.06T + 73T^{2}
79 1+(2.24+1.29i)T+(39.5+68.4i)T2 1 + (2.24 + 1.29i)T + (39.5 + 68.4i)T^{2}
83 1+(3.98+2.30i)T+(41.5+71.8i)T2 1 + (3.98 + 2.30i)T + (41.5 + 71.8i)T^{2}
89 1−8.63iT−89T2 1 - 8.63iT - 89T^{2}
97 1+(−3.35+5.81i)T+(−48.5−84.0i)T2 1 + (-3.35 + 5.81i)T + (-48.5 - 84.0i)T^{2}
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   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

−14.97978302154472165615105386294, −14.45517755311644382171344605744, −12.69496720952127909143360765253, −11.85539473569968757729645079643, −10.58436726241657733162589750484, −8.875515933288566899556320357171, −8.238223075517799309705814316295, −6.31876240618624453363936905294, −5.17062500644679992682256880194, −4.18520632964577858566156115719, 1.62792039028054742034256897381, 4.06632531188568629328634915562, 5.54589142181019492760130056483, 7.21319963037127785582048995754, 8.515452925765671979536433076908, 10.38996401748245505406927314155, 11.34870139642146937120036835624, 11.69542468693414966146258707567, 13.28362393395700364297206153459, 13.93035407755272585140931107040

Graph of the ZZ-function along the critical line