Properties

Label 2-4080-5.4-c1-0-3
Degree 22
Conductor 40804080
Sign i-i
Analytic cond. 32.578932.5789
Root an. cond. 5.707795.70779
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  i·3-s − 2.23i·5-s + 1.23i·7-s − 9-s − 2·11-s − 5.23i·13-s − 2.23·15-s + i·17-s − 6.47·19-s + 1.23·21-s + 4i·23-s − 5.00·25-s + i·27-s + 4·29-s − 2.47·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.999i·5-s + 0.467i·7-s − 0.333·9-s − 0.603·11-s − 1.45i·13-s − 0.577·15-s + 0.242i·17-s − 1.48·19-s + 0.269·21-s + 0.834i·23-s − 1.00·25-s + 0.192i·27-s + 0.742·29-s − 0.444·31-s + ⋯

Functional equation

Λ(s)=(4080s/2ΓC(s)L(s)=(iΛ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4080s/2ΓC(s+1/2)L(s)=(iΛ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40804080    =    2435172^{4} \cdot 3 \cdot 5 \cdot 17
Sign: i-i
Analytic conductor: 32.578932.5789
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4080(2449,)\chi_{4080} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4080, ( :1/2), i)(2,\ 4080,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) \approx 0.28194799590.2819479959
L(12)L(\frac12) \approx 0.28194799590.2819479959
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)
bad2 1 1
3 1+iT 1 + iT
5 1+2.23iT 1 + 2.23iT
17 1iT 1 - iT
good7 11.23iT7T2 1 - 1.23iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+5.23iT13T2 1 + 5.23iT - 13T^{2}
19 1+6.47T+19T2 1 + 6.47T + 19T^{2}
23 14iT23T2 1 - 4iT - 23T^{2}
29 14T+29T2 1 - 4T + 29T^{2}
31 1+2.47T+31T2 1 + 2.47T + 31T^{2}
37 12iT37T2 1 - 2iT - 37T^{2}
41 1+7.70T+41T2 1 + 7.70T + 41T^{2}
43 112.1iT43T2 1 - 12.1iT - 43T^{2}
47 1+10.4iT47T2 1 + 10.4iT - 47T^{2}
53 110.9iT53T2 1 - 10.9iT - 53T^{2}
59 111.7T+59T2 1 - 11.7T + 59T^{2}
61 14.47T+61T2 1 - 4.47T + 61T^{2}
67 11.70iT67T2 1 - 1.70iT - 67T^{2}
71 1+11.2T+71T2 1 + 11.2T + 71T^{2}
73 1+8.76iT73T2 1 + 8.76iT - 73T^{2}
79 10.944T+79T2 1 - 0.944T + 79T^{2}
83 110.4iT83T2 1 - 10.4iT - 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+1.70iT97T2 1 + 1.70iT - 97T^{2}
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   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

−8.356268082747924537089856823767, −8.154217623579506587427544614520, −7.27401757610306078317800045097, −6.28968987205691368975934624584, −5.59072527763094925640632060388, −5.09300375656129953604384451321, −4.10610048722845622415751108265, −3.03120219789291241890020678953, −2.15056542184496250130093119113, −1.10301304985688485936536160564, 0.082846234740168725565072584165, 1.98247643508177530252271915590, 2.67782010055908602708073282445, 3.79966083414469113715176292498, 4.27089181925853376177795912031, 5.17843912155961614641938952091, 6.19242628248492608153194271865, 6.81071349077508861342855292016, 7.32066156544698011912306538269, 8.434058776989840827254572558587

Graph of the ZZ-function along the critical line