↑

Properties

Label 2-1470-35.13-c1-0-18
Degree 22
Conductor 14701470
Sign 0.566+0.824i0.566 + 0.824i
Analytic cond. 11.738011.7380
Root an. cond. 3.426073.42607
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)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−0.569 + 2.16i)5-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (1.12 + 1.93i)10-s − 2.01·11-s + (−0.707 − 0.707i)12-s + (3.44 − 3.44i)13-s + (1.12 + 1.93i)15-s − 1.00·16-s + (3.40 + 3.40i)17-s + (−0.707 − 0.707i)18-s + 6.72·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.254 + 0.966i)5-s − 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.356 + 0.610i)10-s − 0.607·11-s + (−0.204 − 0.204i)12-s + (0.954 − 0.954i)13-s + (0.290 + 0.498i)15-s − 0.250·16-s + (0.824 + 0.824i)17-s + (−0.166 − 0.166i)18-s + 1.54·19-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=((0.566+0.824i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+1/2) L(s)=((0.566+0.824i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: 0.566+0.824i0.566 + 0.824i
Analytic conductor: 11.738011.7380
Root analytic conductor: 3.426073.42607
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1470(1273,⋅)\chi_{1470} (1273, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1470, ( :1/2), 0.566+0.824i)(2,\ 1470,\ (\ :1/2),\ 0.566 + 0.824i)

Particular Values

L(1)L(1) ≈\approx 2.5608199792.560819979
L(12)L(\frac12) ≈\approx 2.5608199792.560819979
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
5 1+(0.569−2.16i)T 1 + (0.569 - 2.16i)T
7 1 1
good11 1+2.01T+11T2 1 + 2.01T + 11T^{2}
13 1+(−3.44+3.44i)T−13iT2 1 + (-3.44 + 3.44i)T - 13iT^{2}
17 1+(−3.40−3.40i)T+17iT2 1 + (-3.40 - 3.40i)T + 17iT^{2}
19 1−6.72T+19T2 1 - 6.72T + 19T^{2}
23 1+(−4.65−4.65i)T+23iT2 1 + (-4.65 - 4.65i)T + 23iT^{2}
29 1+1.60iT−29T2 1 + 1.60iT - 29T^{2}
31 1+6.25iT−31T2 1 + 6.25iT - 31T^{2}
37 1+(−3.63+3.63i)T−37iT2 1 + (-3.63 + 3.63i)T - 37iT^{2}
41 1+4.94iT−41T2 1 + 4.94iT - 41T^{2}
43 1+(4.68+4.68i)T+43iT2 1 + (4.68 + 4.68i)T + 43iT^{2}
47 1+(1.07+1.07i)T+47iT2 1 + (1.07 + 1.07i)T + 47iT^{2}
53 1+(−1.42−1.42i)T+53iT2 1 + (-1.42 - 1.42i)T + 53iT^{2}
59 1−12.8T+59T2 1 - 12.8T + 59T^{2}
61 1−9.00iT−61T2 1 - 9.00iT - 61T^{2}
67 1+(1.19−1.19i)T−67iT2 1 + (1.19 - 1.19i)T - 67iT^{2}
71 1+3.49T+71T2 1 + 3.49T + 71T^{2}
73 1+(−5.97+5.97i)T−73iT2 1 + (-5.97 + 5.97i)T - 73iT^{2}
79 1−13.9iT−79T2 1 - 13.9iT - 79T^{2}
83 1+(8.59−8.59i)T−83iT2 1 + (8.59 - 8.59i)T - 83iT^{2}
89 1−2.40T+89T2 1 - 2.40T + 89T^{2}
97 1+(−13.1−13.1i)T+97iT2 1 + (-13.1 - 13.1i)T + 97iT^{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

−9.637941583523398149233672165985, −8.448126967310616644609545433165, −7.67970823009817018051282933701, −7.08734424030142232994662221191, −5.86748239533610004411432674689, −5.44811850078426686570098063400, −3.80856244597964739009220702146, −3.32227081024154629523415582018, −2.43705960127092727274448564975, −1.04310209863073920489897138161, 1.21595988038675845777326487127, 2.91136328930635370463895934832, 3.69924421119195664785603844940, 4.91394682550245085418666556702, 5.06509935523403313776685365120, 6.31189675464645731316448288335, 7.30514602714017926112875974396, 8.051611305300310347214899715820, 8.775926910776084425487752076013, 9.382664515039381557342395399866

Graph of the ZZ-function along the critical line