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Properties

Label 2-930-5.4-c1-0-6
Degree 22
Conductor 930930
Sign 0.165−0.986i0.165 - 0.986i
Analytic cond. 7.426087.42608
Root an. cond. 2.725082.72508
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·2-s + i·3-s − 4-s + (−0.369 + 2.20i)5-s + 6-s − 2i·7-s + i·8-s − 9-s + (2.20 + 0.369i)10-s + 3.26·11-s − i·12-s + 2.73i·13-s − 2·14-s + (−2.20 − 0.369i)15-s + 16-s + 4.93i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.165 + 0.986i)5-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (0.697 + 0.116i)10-s + 0.983·11-s − 0.288i·12-s + 0.759i·13-s − 0.534·14-s + (−0.569 − 0.0954i)15-s + 0.250·16-s + 1.19i·17-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((0.165−0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((0.165−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: 0.165−0.986i0.165 - 0.986i
Analytic conductor: 7.426087.42608
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ930(559,⋅)\chi_{930} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 930, ( :1/2), 0.165−0.986i)(2,\ 930,\ (\ :1/2),\ 0.165 - 0.986i)

Particular Values

L(1)L(1) ≈\approx 0.855363+0.723923i0.855363 + 0.723923i
L(12)L(\frac12) ≈\approx 0.855363+0.723923i0.855363 + 0.723923i
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+iT 1 + iT
3 1−iT 1 - iT
5 1+(0.369−2.20i)T 1 + (0.369 - 2.20i)T
31 1+T 1 + T
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1−3.26T+11T2 1 - 3.26T + 11T^{2}
13 1−2.73iT−13T2 1 - 2.73iT - 13T^{2}
17 1−4.93iT−17T2 1 - 4.93iT - 17T^{2}
19 1+4.93T+19T2 1 + 4.93T + 19T^{2}
23 1+0.521iT−23T2 1 + 0.521iT - 23T^{2}
29 1+0.521T+29T2 1 + 0.521T + 29T^{2}
37 1−10.8iT−37T2 1 - 10.8iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−8.82iT−43T2 1 - 8.82iT - 43T^{2}
47 1+4.93iT−47T2 1 + 4.93iT - 47T^{2}
53 1−13.3iT−53T2 1 - 13.3iT - 53T^{2}
59 1+9.34T+59T2 1 + 9.34T + 59T^{2}
61 1+9.75T+61T2 1 + 9.75T + 61T^{2}
67 1−13.5iT−67T2 1 - 13.5iT - 67T^{2}
71 1−5.56T+71T2 1 - 5.56T + 71T^{2}
73 1+3.04iT−73T2 1 + 3.04iT - 73T^{2}
79 1−6.41T+79T2 1 - 6.41T + 79T^{2}
83 1+1.36iT−83T2 1 + 1.36iT - 83T^{2}
89 1−11.8T+89T2 1 - 11.8T + 89T^{2}
97 1−7.26iT−97T2 1 - 7.26iT - 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

−10.46655825348802805256756301066, −9.641108318083680428172802930666, −8.797063629765069323012637022166, −7.85142825493578631508749918709, −6.70218853915673204038048057002, −6.09406640304293341632189353782, −4.39630658267190707424944343763, −4.02776596307716053345255355874, −2.99862139772311652847097024544, −1.63117750483127575910222366453, 0.53985296122424655334859698180, 2.09185019993051032073452075394, 3.66575857188705674443097216072, 4.82320216324444104996066113937, 5.61558989041235589979453314184, 6.40299417236218662198232875328, 7.41937040255015665357575594626, 8.153636414498025660664739553498, 9.086657328676041685426707427068, 9.278107629284690472144276525425

Graph of the ZZ-function along the critical line