Properties

Label 2-735-105.104-c1-0-14
Degree 22
Conductor 735735
Sign 0.9750.221i0.975 - 0.221i
Analytic cond. 5.869005.86900
Root an. cond. 2.422602.42260
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  − 1.51·2-s + (−1.66 − 0.476i)3-s + 0.294·4-s + (0.775 − 2.09i)5-s + (2.52 + 0.722i)6-s + 2.58·8-s + (2.54 + 1.58i)9-s + (−1.17 + 3.17i)10-s + 2.14i·11-s + (−0.490 − 0.140i)12-s − 3.48·13-s + (−2.29 + 3.12i)15-s − 4.50·16-s + 3.57i·17-s + (−3.85 − 2.40i)18-s − 1.22i·19-s + ⋯
L(s)  = 1  − 1.07·2-s + (−0.961 − 0.275i)3-s + 0.147·4-s + (0.346 − 0.937i)5-s + (1.02 + 0.294i)6-s + 0.913·8-s + (0.848 + 0.529i)9-s + (−0.371 + 1.00i)10-s + 0.647i·11-s + (−0.141 − 0.0405i)12-s − 0.965·13-s + (−0.591 + 0.806i)15-s − 1.12·16-s + 0.867i·17-s + (−0.908 − 0.566i)18-s − 0.280i·19-s + ⋯

Functional equation

Λ(s)=(735s/2ΓC(s)L(s)=((0.9750.221i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2ΓC(s+1/2)L(s)=((0.9750.221i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    35723 \cdot 5 \cdot 7^{2}
Sign: 0.9750.221i0.975 - 0.221i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(734,)\chi_{735} (734, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), 0.9750.221i)(2,\ 735,\ (\ :1/2),\ 0.975 - 0.221i)

Particular Values

L(1)L(1) \approx 0.500436+0.0562466i0.500436 + 0.0562466i
L(12)L(\frac12) \approx 0.500436+0.0562466i0.500436 + 0.0562466i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(1.66+0.476i)T 1 + (1.66 + 0.476i)T
5 1+(0.775+2.09i)T 1 + (-0.775 + 2.09i)T
7 1 1
good2 1+1.51T+2T2 1 + 1.51T + 2T^{2}
11 12.14iT11T2 1 - 2.14iT - 11T^{2}
13 1+3.48T+13T2 1 + 3.48T + 13T^{2}
17 13.57iT17T2 1 - 3.57iT - 17T^{2}
19 1+1.22iT19T2 1 + 1.22iT - 19T^{2}
23 11.51T+23T2 1 - 1.51T + 23T^{2}
29 15.95iT29T2 1 - 5.95iT - 29T^{2}
31 1+3.17iT31T2 1 + 3.17iT - 31T^{2}
37 17.80iT37T2 1 - 7.80iT - 37T^{2}
41 111.8T+41T2 1 - 11.8T + 41T^{2}
43 12.99iT43T2 1 - 2.99iT - 43T^{2}
47 16.10iT47T2 1 - 6.10iT - 47T^{2}
53 111.2T+53T2 1 - 11.2T + 53T^{2}
59 1+2.16T+59T2 1 + 2.16T + 59T^{2}
61 1+3.39iT61T2 1 + 3.39iT - 61T^{2}
67 110.3iT67T2 1 - 10.3iT - 67T^{2}
71 1+10.3iT71T2 1 + 10.3iT - 71T^{2}
73 16.85T+73T2 1 - 6.85T + 73T^{2}
79 1+1.88T+79T2 1 + 1.88T + 79T^{2}
83 1+9.10iT83T2 1 + 9.10iT - 83T^{2}
89 11.77T+89T2 1 - 1.77T + 89T^{2}
97 11.32T+97T2 1 - 1.32T + 97T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−10.20195822778352397853823661247, −9.630888394188933091757836218470, −8.826538605411777022506774960746, −7.83986423520135101495408966361, −7.16334546629806346446861757023, −6.03218304013966704791314865584, −4.94303605644368709404316992082, −4.40253254363522616379739570003, −2.02674578096365277978578099842, −0.934157258717622031828074566707, 0.58413804265747357271270940679, 2.33943488734371595700592440606, 3.90642701935412226777154981957, 5.09101692645056914478478046596, 6.00054361514303294328229248228, 7.09619458023411752185073348117, 7.57143344896631631514888219502, 8.918102659414848953927303949689, 9.686775526056532084350634735722, 10.23542909689979247025013858767

Graph of the ZZ-function along the critical line