↑

Properties

Label 2-2156-7.4-c1-0-13
Degree 22
Conductor 21562156
Sign 0.991−0.126i0.991 - 0.126i
Analytic cond. 17.215717.2157
Root an. cond. 4.149184.14918
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 − 1.22i)3-s + (0.500 + 0.866i)9-s + (0.5 − 0.866i)11-s + 1.41·13-s + (−3.53 + 6.12i)17-s + (1.41 + 2.44i)19-s + (2 + 3.46i)23-s + (2.5 − 4.33i)25-s + 5.65·27-s + 2·29-s + (−2.12 + 3.67i)31-s + (−0.707 − 1.22i)33-s + (2 + 3.46i)37-s + (1.00 − 1.73i)39-s − 1.41·41-s + ⋯
L(s)  = 1  + (0.408 − 0.707i)3-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s + 0.392·13-s + (−0.857 + 1.48i)17-s + (0.324 + 0.561i)19-s + (0.417 + 0.722i)23-s + (0.5 − 0.866i)25-s + 1.08·27-s + 0.371·29-s + (−0.381 + 0.659i)31-s + (−0.123 − 0.213i)33-s + (0.328 + 0.569i)37-s + (0.160 − 0.277i)39-s − 0.220·41-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.991−0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s+1/2) L(s)=((0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.991−0.126i0.991 - 0.126i
Analytic conductor: 17.215717.2157
Root analytic conductor: 4.149184.14918
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2156(1145,⋅)\chi_{2156} (1145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :1/2), 0.991−0.126i)(2,\ 2156,\ (\ :1/2),\ 0.991 - 0.126i)

Particular Values

L(1)L(1) ≈\approx 2.1175465902.117546590
L(12)L(\frac12) ≈\approx 2.1175465902.117546590
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
11 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(−0.707+1.22i)T+(−1.5−2.59i)T2 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2}
5 1+(−2.5+4.33i)T2 1 + (-2.5 + 4.33i)T^{2}
13 1−1.41T+13T2 1 - 1.41T + 13T^{2}
17 1+(3.53−6.12i)T+(−8.5−14.7i)T2 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.41−2.44i)T+(−9.5+16.4i)T2 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2−3.46i)T+(−11.5+19.9i)T2 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+(2.12−3.67i)T+(−15.5−26.8i)T2 1 + (2.12 - 3.67i)T + (-15.5 - 26.8i)T^{2}
37 1+(−2−3.46i)T+(−18.5+32.0i)T2 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+1.41T+41T2 1 + 1.41T + 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1+(−4.94−8.57i)T+(−23.5+40.7i)T2 1 + (-4.94 - 8.57i)T + (-23.5 + 40.7i)T^{2}
53 1+(−2+3.46i)T+(−26.5−45.8i)T2 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.12+3.67i)T+(−29.5−51.0i)T2 1 + (-2.12 + 3.67i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.36+11.0i)T+(−30.5+52.8i)T2 1 + (6.36 + 11.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4+6.92i)T+(−33.5−58.0i)T2 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1+(−0.707+1.22i)T+(−36.5−63.2i)T2 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5−8.66i)T+(−39.5+68.4i)T2 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2}
83 1+8.48T+83T2 1 + 8.48T + 83T^{2}
89 1+(−5.65−9.79i)T+(−44.5+77.0i)T2 1 + (-5.65 - 9.79i)T + (-44.5 + 77.0i)T^{2}
97 1+8.48T+97T2 1 + 8.48T + 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

−8.846512966559789157660729230671, −8.271412877891261898249201306546, −7.68306485728899263419824817733, −6.69999265539355252856615103271, −6.19913271860185442274621953720, −5.10263312365293997878856193351, −4.13894262629125312046146407048, −3.18501656123145836384266699957, −2.07962606377655548064038089994, −1.20921582178916836626915820982, 0.810889429789893705305067641421, 2.41499464965090300152443686101, 3.24532167763463205435077415999, 4.24742316397698469615886274129, 4.82547367399436220357001523186, 5.83802496510041860472718140662, 6.96860134462120088880204811471, 7.28852330944209207686613053924, 8.704789028196061167402114690447, 8.992951849696607716599342185315

Graph of the ZZ-function along the critical line