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Properties

Label 2-538-269.268-c1-0-14
Degree 22
Conductor 538538
Sign 0.0996+0.995i0.0996 + 0.995i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
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 − 1.53i·3-s − 4-s + 3.51·5-s − 1.53·6-s + 1.46i·7-s + i·8-s + 0.641·9-s − 3.51i·10-s + 2.87·11-s + 1.53i·12-s + 1.66·13-s + 1.46·14-s − 5.40i·15-s + 16-s − 0.593i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.886i·3-s − 0.5·4-s + 1.57·5-s − 0.626·6-s + 0.553i·7-s + 0.353i·8-s + 0.213·9-s − 1.11i·10-s + 0.865·11-s + 0.443i·12-s + 0.460·13-s + 0.391·14-s − 1.39i·15-s + 0.250·16-s − 0.143i·17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.0996+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.0996+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.0996+0.995i0.0996 + 0.995i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(537,⋅)\chi_{538} (537, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.0996+0.995i)(2,\ 538,\ (\ :1/2),\ 0.0996 + 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.41291−1.27854i1.41291 - 1.27854i
L(12)L(\frac12) ≈\approx 1.41291−1.27854i1.41291 - 1.27854i
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
269 1+(−16.3+1.63i)T 1 + (-16.3 + 1.63i)T
good3 1+1.53iT−3T2 1 + 1.53iT - 3T^{2}
5 1−3.51T+5T2 1 - 3.51T + 5T^{2}
7 1−1.46iT−7T2 1 - 1.46iT - 7T^{2}
11 1−2.87T+11T2 1 - 2.87T + 11T^{2}
13 1−1.66T+13T2 1 - 1.66T + 13T^{2}
17 1+0.593iT−17T2 1 + 0.593iT - 17T^{2}
19 1−4.67iT−19T2 1 - 4.67iT - 19T^{2}
23 1+3.91T+23T2 1 + 3.91T + 23T^{2}
29 1+6.42iT−29T2 1 + 6.42iT - 29T^{2}
31 1−7.26iT−31T2 1 - 7.26iT - 31T^{2}
37 1+9.91T+37T2 1 + 9.91T + 37T^{2}
41 1+4.88T+41T2 1 + 4.88T + 41T^{2}
43 1+12.5T+43T2 1 + 12.5T + 43T^{2}
47 1+5.42T+47T2 1 + 5.42T + 47T^{2}
53 1−5.55T+53T2 1 - 5.55T + 53T^{2}
59 1+11.1iT−59T2 1 + 11.1iT - 59T^{2}
61 1+4.34T+61T2 1 + 4.34T + 61T^{2}
67 1+6.51T+67T2 1 + 6.51T + 67T^{2}
71 1+8.19iT−71T2 1 + 8.19iT - 71T^{2}
73 1−1.03T+73T2 1 - 1.03T + 73T^{2}
79 1−4.28T+79T2 1 - 4.28T + 79T^{2}
83 1−16.7iT−83T2 1 - 16.7iT - 83T^{2}
89 1−6.07T+89T2 1 - 6.07T + 89T^{2}
97 1−12.2T+97T2 1 - 12.2T + 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

−10.37938810237226794935334656219, −9.907775375249318918737801875358, −8.985486193702039436270796183492, −8.147643554941743646842793950155, −6.69877519558436893027844618275, −6.11807041532103573916434014428, −5.10010505608381989486651356053, −3.55404846017821729785672213438, −2.03408527579213566819331891315, −1.52660865951433510631966602581, 1.62219150454502880701789499477, 3.50995385223220903134250914850, 4.58070784953288673384181642523, 5.47099923174319052689621031271, 6.44595198047561979362505806061, 7.14524400852132381995797635590, 8.703179281467601625254239522040, 9.249081290331785530193773807641, 10.12018606574832443062530698080, 10.51644031564366691109144307225

Graph of the ZZ-function along the critical line