↑

Properties

Label 2-192-16.13-c1-0-3
Degree 22
Conductor 192192
Sign 0.513+0.857i0.513 + 0.857i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
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  + (−0.707 − 0.707i)3-s + (1.27 − 1.27i)5-s − 0.158i·7-s + 1.00i·9-s + (3.79 − 3.79i)11-s + (−4.21 − 4.21i)13-s − 1.79·15-s + 3.05·17-s + (2.15 + 2.15i)19-s + (−0.112 + 0.112i)21-s + 2.82i·23-s + 1.76i·25-s + (0.707 − 0.707i)27-s + (2.09 + 2.09i)29-s − 4.15·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.568 − 0.568i)5-s − 0.0600i·7-s + 0.333i·9-s + (1.14 − 1.14i)11-s + (−1.16 − 1.16i)13-s − 0.464·15-s + 0.740·17-s + (0.495 + 0.495i)19-s + (−0.0245 + 0.0245i)21-s + 0.589i·23-s + 0.353i·25-s + (0.136 − 0.136i)27-s + (0.389 + 0.389i)29-s − 0.746·31-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=((0.513+0.857i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+1/2) L(s)=((0.513+0.857i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: 0.513+0.857i0.513 + 0.857i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ192(145,⋅)\chi_{192} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :1/2), 0.513+0.857i)(2,\ 192,\ (\ :1/2),\ 0.513 + 0.857i)

Particular Values

L(1)L(1) ≈\approx 0.988890−0.560387i0.988890 - 0.560387i
L(12)L(\frac12) ≈\approx 0.988890−0.560387i0.988890 - 0.560387i
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
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good5 1+(−1.27+1.27i)T−5iT2 1 + (-1.27 + 1.27i)T - 5iT^{2}
7 1+0.158iT−7T2 1 + 0.158iT - 7T^{2}
11 1+(−3.79+3.79i)T−11iT2 1 + (-3.79 + 3.79i)T - 11iT^{2}
13 1+(4.21+4.21i)T+13iT2 1 + (4.21 + 4.21i)T + 13iT^{2}
17 1−3.05T+17T2 1 - 3.05T + 17T^{2}
19 1+(−2.15−2.15i)T+19iT2 1 + (-2.15 - 2.15i)T + 19iT^{2}
23 1−2.82iT−23T2 1 - 2.82iT - 23T^{2}
29 1+(−2.09−2.09i)T+29iT2 1 + (-2.09 - 2.09i)T + 29iT^{2}
31 1+4.15T+31T2 1 + 4.15T + 31T^{2}
37 1+(5.98−5.98i)T−37iT2 1 + (5.98 - 5.98i)T - 37iT^{2}
41 1−2.60iT−41T2 1 - 2.60iT - 41T^{2}
43 1+(5.75−5.75i)T−43iT2 1 + (5.75 - 5.75i)T - 43iT^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+(−3.55+3.55i)T−53iT2 1 + (-3.55 + 3.55i)T - 53iT^{2}
59 1+(4−4i)T−59iT2 1 + (4 - 4i)T - 59iT^{2}
61 1+(−3.66−3.66i)T+61iT2 1 + (-3.66 - 3.66i)T + 61iT^{2}
67 1+(0.767+0.767i)T+67iT2 1 + (0.767 + 0.767i)T + 67iT^{2}
71 1−0.317iT−71T2 1 - 0.317iT - 71T^{2}
73 1−1.33iT−73T2 1 - 1.33iT - 73T^{2}
79 1−9.69T+79T2 1 - 9.69T + 79T^{2}
83 1+(0.115+0.115i)T+83iT2 1 + (0.115 + 0.115i)T + 83iT^{2}
89 1+14.3iT−89T2 1 + 14.3iT - 89T^{2}
97 1+0.571T+97T2 1 + 0.571T + 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

−12.32427035096230560803436669269, −11.62323711111200958754092101823, −10.36889773308737704783543982604, −9.447744822486151092377390517691, −8.322741191113076581470058029429, −7.20898993143409779916004574814, −5.88520125345707706699712368694, −5.17148727733757249257006093125, −3.30929555236875743010133731146, −1.24264254162198050576480100326, 2.17333486573467991237480334359, 4.03125249694446395002958160784, 5.18640800857765897411880961294, 6.58402442750176547034362222669, 7.23679079770375296003894875878, 9.099073856968484862542039367852, 9.746443733095670553139433762766, 10.60063098243362821913589421191, 11.92860753412365095676112117674, 12.28394846118638302003926688258

Graph of the ZZ-function along the critical line