↑

Properties

Label 2-435-145.144-c1-0-3
Degree 22
Conductor 435435
Sign 0.362−0.931i0.362 - 0.931i
Analytic cond. 3.473493.47349
Root an. cond. 1.863731.86373
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  − 1.60·2-s + 3-s + 0.574·4-s + (−2.22 + 0.232i)5-s − 1.60·6-s + 1.04i·7-s + 2.28·8-s + 9-s + (3.56 − 0.373i)10-s − 5.71i·11-s + 0.574·12-s + 4.68i·13-s − 1.66i·14-s + (−2.22 + 0.232i)15-s − 4.81·16-s + 2.22·17-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.287·4-s + (−0.994 + 0.104i)5-s − 0.655·6-s + 0.393i·7-s + 0.808·8-s + 0.333·9-s + (1.12 − 0.118i)10-s − 1.72i·11-s + 0.165·12-s + 1.30i·13-s − 0.446i·14-s + (−0.574 + 0.0601i)15-s − 1.20·16-s + 0.540·17-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((0.362−0.931i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((0.362−0.931i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: 0.362−0.931i0.362 - 0.931i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(289,⋅)\chi_{435} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.362−0.931i)(2,\ 435,\ (\ :1/2),\ 0.362 - 0.931i)

Particular Values

L(1)L(1) ≈\approx 0.538163+0.367960i0.538163 + 0.367960i
L(12)L(\frac12) ≈\approx 0.538163+0.367960i0.538163 + 0.367960i
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)
bad3 1−T 1 - T
5 1+(2.22−0.232i)T 1 + (2.22 - 0.232i)T
29 1+(2.46−4.78i)T 1 + (2.46 - 4.78i)T
good2 1+1.60T+2T2 1 + 1.60T + 2T^{2}
7 1−1.04iT−7T2 1 - 1.04iT - 7T^{2}
11 1+5.71iT−11T2 1 + 5.71iT - 11T^{2}
13 1−4.68iT−13T2 1 - 4.68iT - 13T^{2}
17 1−2.22T+17T2 1 - 2.22T + 17T^{2}
19 1−6.45iT−19T2 1 - 6.45iT - 19T^{2}
23 1−6.19iT−23T2 1 - 6.19iT - 23T^{2}
31 1−2.73iT−31T2 1 - 2.73iT - 31T^{2}
37 1−7.69T+37T2 1 - 7.69T + 37T^{2}
41 1−2.07iT−41T2 1 - 2.07iT - 41T^{2}
43 1−0.0721T+43T2 1 - 0.0721T + 43T^{2}
47 1+3.86T+47T2 1 + 3.86T + 47T^{2}
53 1−10.2iT−53T2 1 - 10.2iT - 53T^{2}
59 1−9.67T+59T2 1 - 9.67T + 59T^{2}
61 1+5.68iT−61T2 1 + 5.68iT - 61T^{2}
67 1−8.51iT−67T2 1 - 8.51iT - 67T^{2}
71 1+9.26T+71T2 1 + 9.26T + 71T^{2}
73 1+9.65T+73T2 1 + 9.65T + 73T^{2}
79 1−1.03iT−79T2 1 - 1.03iT - 79T^{2}
83 1−1.94iT−83T2 1 - 1.94iT - 83T^{2}
89 1+16.4iT−89T2 1 + 16.4iT - 89T^{2}
97 1−1.12T+97T2 1 - 1.12T + 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

−11.24869530433448777204430292965, −10.25275089126027312428162722165, −9.223086915442980229426353337768, −8.623520737728335700659191823041, −7.948424647321346184809625963853, −7.16009073077967466686187376537, −5.76880923190783678204074329229, −4.19036805946356723940319472916, −3.24047970887480679852681138463, −1.37797590424918681095652665613, 0.64093105730469483870008168632, 2.49059251847074004020124858044, 4.09080023013695440289806481023, 4.86811452509713996987268291629, 6.93191317814994660038322608926, 7.60516535799834749076755397305, 8.160836174850975606635247122796, 9.135854474936574864693556159445, 9.976667654668664173127068478770, 10.60488252500010425991590545513

Graph of the ZZ-function along the critical line