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Properties

Label 2-435-145.144-c1-0-23
Degree 22
Conductor 435435
Sign 0.0830+0.996i0.0830 + 0.996i
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + (2 − i)5-s − 2·6-s − 4i·7-s + 9-s + (−4 + 2i)10-s − i·11-s + 2·12-s − 2i·13-s + 8i·14-s + (2 − i)15-s − 4·16-s − 6·17-s − 2·18-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + (0.894 − 0.447i)5-s − 0.816·6-s − 1.51i·7-s + 0.333·9-s + (−1.26 + 0.632i)10-s − 0.301i·11-s + 0.577·12-s − 0.554i·13-s + 2.13i·14-s + (0.516 − 0.258i)15-s − 16-s − 1.45·17-s − 0.471·18-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((0.0830+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((0.0830+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: 0.0830+0.996i0.0830 + 0.996i
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), 0.0830+0.996i)(2,\ 435,\ (\ :1/2),\ 0.0830 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.645544−0.593986i0.645544 - 0.593986i
L(12)L(\frac12) ≈\approx 0.645544−0.593986i0.645544 - 0.593986i
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+i)T 1 + (-2 + i)T
29 1+(2−5i)T 1 + (2 - 5i)T
good2 1+2T+2T2 1 + 2T + 2T^{2}
7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1+iT−11T2 1 + iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1−9iT−23T2 1 - 9iT - 23T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+9iT−41T2 1 + 9iT - 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1+9iT−53T2 1 + 9iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−15T+73T2 1 - 15T + 73T^{2}
79 1−4iT−79T2 1 - 4iT - 79T^{2}
83 1−7iT−83T2 1 - 7iT - 83T^{2}
89 1−2iT−89T2 1 - 2iT - 89T^{2}
97 1−11T+97T2 1 - 11T + 97T^{2}
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   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

−10.64376853280538732873835196580, −9.825559453686683173643219770104, −9.158389456161346471318345142439, −8.435083951764939988995447218328, −7.38245951446731992835070226902, −6.79568999643739799447282039592, −5.15598333046146503472668996480, −3.85144277930253994974000303287, −2.11783780223542915033328418052, −0.834978617783562150213868522314, 1.96327910702602339931425882472, 2.52509541441722889404633038325, 4.57114576488874229321384665906, 6.09095042061882719816334988417, 6.82866517287244109916693696737, 8.095690002938542576854341536020, 8.875556592568695344666217360988, 9.307911693400769796865292465243, 10.16611440680651221428335497561, 10.97354914286742907840547048921

Graph of the ZZ-function along the critical line