↑

Properties

Label 2-54-27.25-c1-0-1
Degree 22
Conductor 5454
Sign 0.943+0.331i0.943 + 0.331i
Analytic cond. 0.4311920.431192
Root an. cond. 0.6566520.656652
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  + (0.173 − 0.984i)2-s + (0.140 + 1.72i)3-s + (−0.939 − 0.342i)4-s + (2.42 − 2.03i)5-s + (1.72 + 0.161i)6-s + (−3.46 + 1.26i)7-s + (−0.5 + 0.866i)8-s + (−2.96 + 0.484i)9-s + (−1.58 − 2.74i)10-s + (−1.75 − 1.46i)11-s + (0.458 − 1.67i)12-s + (0.538 + 3.05i)13-s + (0.640 + 3.62i)14-s + (3.85 + 3.90i)15-s + (0.766 + 0.642i)16-s + (−0.862 − 1.49i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.0810 + 0.996i)3-s + (−0.469 − 0.171i)4-s + (1.08 − 0.910i)5-s + (0.704 + 0.0659i)6-s + (−1.30 + 0.476i)7-s + (−0.176 + 0.306i)8-s + (−0.986 + 0.161i)9-s + (−0.500 − 0.867i)10-s + (−0.527 − 0.442i)11-s + (0.132 − 0.482i)12-s + (0.149 + 0.846i)13-s + (0.171 + 0.970i)14-s + (0.995 + 1.00i)15-s + (0.191 + 0.160i)16-s + (−0.209 − 0.362i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.943+0.331i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1/2) L(s)=((0.943+0.331i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.943+0.331i0.943 + 0.331i
Analytic conductor: 0.4311920.431192
Root analytic conductor: 0.6566520.656652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ54(25,⋅)\chi_{54} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1/2), 0.943+0.331i)(2,\ 54,\ (\ :1/2),\ 0.943 + 0.331i)

Particular Values

L(1)L(1) ≈\approx 0.879971−0.150256i0.879971 - 0.150256i
L(12)L(\frac12) ≈\approx 0.879971−0.150256i0.879971 - 0.150256i
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+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
3 1+(−0.140−1.72i)T 1 + (-0.140 - 1.72i)T
good5 1+(−2.42+2.03i)T+(0.868−4.92i)T2 1 + (-2.42 + 2.03i)T + (0.868 - 4.92i)T^{2}
7 1+(3.46−1.26i)T+(5.36−4.49i)T2 1 + (3.46 - 1.26i)T + (5.36 - 4.49i)T^{2}
11 1+(1.75+1.46i)T+(1.91+10.8i)T2 1 + (1.75 + 1.46i)T + (1.91 + 10.8i)T^{2}
13 1+(−0.538−3.05i)T+(−12.2+4.44i)T2 1 + (-0.538 - 3.05i)T + (-12.2 + 4.44i)T^{2}
17 1+(0.862+1.49i)T+(−8.5+14.7i)T2 1 + (0.862 + 1.49i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.69+2.93i)T+(−9.5−16.4i)T2 1 + (-1.69 + 2.93i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.15−1.14i)T+(17.6+14.7i)T2 1 + (-3.15 - 1.14i)T + (17.6 + 14.7i)T^{2}
29 1+(0.101−0.576i)T+(−27.2−9.91i)T2 1 + (0.101 - 0.576i)T + (-27.2 - 9.91i)T^{2}
31 1+(−4.35−1.58i)T+(23.7+19.9i)T2 1 + (-4.35 - 1.58i)T + (23.7 + 19.9i)T^{2}
37 1+(−3.65−6.32i)T+(−18.5+32.0i)T2 1 + (-3.65 - 6.32i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.22+6.97i)T+(−38.5+14.0i)T2 1 + (1.22 + 6.97i)T + (-38.5 + 14.0i)T^{2}
43 1+(−1.27−1.06i)T+(7.46+42.3i)T2 1 + (-1.27 - 1.06i)T + (7.46 + 42.3i)T^{2}
47 1+(−3.61+1.31i)T+(36.0−30.2i)T2 1 + (-3.61 + 1.31i)T + (36.0 - 30.2i)T^{2}
53 1+2.58T+53T2 1 + 2.58T + 53T^{2}
59 1+(7.40−6.21i)T+(10.2−58.1i)T2 1 + (7.40 - 6.21i)T + (10.2 - 58.1i)T^{2}
61 1+(12.3−4.47i)T+(46.7−39.2i)T2 1 + (12.3 - 4.47i)T + (46.7 - 39.2i)T^{2}
67 1+(1.49+8.47i)T+(−62.9+22.9i)T2 1 + (1.49 + 8.47i)T + (-62.9 + 22.9i)T^{2}
71 1+(0.993+1.72i)T+(−35.5+61.4i)T2 1 + (0.993 + 1.72i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.32−9.22i)T+(−36.5−63.2i)T2 1 + (5.32 - 9.22i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.44+13.8i)T+(−74.2−27.0i)T2 1 + (-2.44 + 13.8i)T + (-74.2 - 27.0i)T^{2}
83 1+(0.538−3.05i)T+(−77.9−28.3i)T2 1 + (0.538 - 3.05i)T + (-77.9 - 28.3i)T^{2}
89 1+(−8.67+15.0i)T+(−44.5−77.0i)T2 1 + (-8.67 + 15.0i)T + (-44.5 - 77.0i)T^{2}
97 1+(−7.04−5.90i)T+(16.8+95.5i)T2 1 + (-7.04 - 5.90i)T + (16.8 + 95.5i)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

−15.49676549349205841562086732077, −13.82015952008156761400075874094, −13.21040081723104218635661320721, −11.89012543446478273232208449329, −10.45998674048843389824743834186, −9.356336908891079925551746874955, −8.981264507000523657803178351288, −6.03483894661563278655031822731, −4.83378555322537888982443692768, −2.91630507369118712360623845274, 2.95074388022841310194526929879, 5.86887439080793451778033588558, 6.63577943667956578778559378443, 7.76091571887258369058251547947, 9.530707812889934449300478142768, 10.58785746035975508297960593748, 12.63598743506795851661650962670, 13.29752534848387743154550918860, 14.15353192393420481459468555389, 15.28588607784934365085366782655

Graph of the ZZ-function along the critical line