Properties

Label 2-115-115.114-c4-0-45
Degree 22
Conductor 115115
Sign 0.8330.552i0.833 - 0.552i
Analytic cond. 11.887511.8875
Root an. cond. 3.447833.44783
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.76i·2-s − 12.7i·3-s − 29.8·4-s + (−13.2 + 21.2i)5-s − 86.4·6-s + 5.97·7-s + 93.4i·8-s − 82.1·9-s + (143. + 89.3i)10-s + 18.7i·11-s + 380. i·12-s + 30.6i·13-s − 40.4i·14-s + (271. + 168. i)15-s + 155.·16-s − 280.·17-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.41i·3-s − 1.86·4-s + (−0.528 + 0.849i)5-s − 2.40·6-s + 0.121·7-s + 1.46i·8-s − 1.01·9-s + (1.43 + 0.893i)10-s + 0.155i·11-s + 2.64i·12-s + 0.181i·13-s − 0.206i·14-s + (1.20 + 0.749i)15-s + 0.608·16-s − 0.971·17-s + ⋯

Functional equation

Λ(s)=(115s/2ΓC(s)L(s)=((0.8330.552i)Λ(5s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(115s/2ΓC(s+2)L(s)=((0.8330.552i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5235 \cdot 23
Sign: 0.8330.552i0.833 - 0.552i
Analytic conductor: 11.887511.8875
Root analytic conductor: 3.447833.44783
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ115(114,)\chi_{115} (114, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :2), 0.8330.552i)(2,\ 115,\ (\ :2),\ 0.833 - 0.552i)

Particular Values

L(52)L(\frac{5}{2}) \approx 0.265094+0.0799513i0.265094 + 0.0799513i
L(12)L(\frac12) \approx 0.265094+0.0799513i0.265094 + 0.0799513i
L(3)L(3) 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)
bad5 1+(13.221.2i)T 1 + (13.2 - 21.2i)T
23 1+(481.+219.i)T 1 + (481. + 219. i)T
good2 1+6.76iT16T2 1 + 6.76iT - 16T^{2}
3 1+12.7iT81T2 1 + 12.7iT - 81T^{2}
7 15.97T+2.40e3T2 1 - 5.97T + 2.40e3T^{2}
11 118.7iT1.46e4T2 1 - 18.7iT - 1.46e4T^{2}
13 130.6iT2.85e4T2 1 - 30.6iT - 2.85e4T^{2}
17 1+280.T+8.35e4T2 1 + 280.T + 8.35e4T^{2}
19 193.6iT1.30e5T2 1 - 93.6iT - 1.30e5T^{2}
29 1378.T+7.07e5T2 1 - 378.T + 7.07e5T^{2}
31 125.0T+9.23e5T2 1 - 25.0T + 9.23e5T^{2}
37 1+2.50e3T+1.87e6T2 1 + 2.50e3T + 1.87e6T^{2}
41 1182.T+2.82e6T2 1 - 182.T + 2.82e6T^{2}
43 13.03e3T+3.41e6T2 1 - 3.03e3T + 3.41e6T^{2}
47 12.99e3iT4.87e6T2 1 - 2.99e3iT - 4.87e6T^{2}
53 13.12e3T+7.89e6T2 1 - 3.12e3T + 7.89e6T^{2}
59 1+495.T+1.21e7T2 1 + 495.T + 1.21e7T^{2}
61 1+3.23e3iT1.38e7T2 1 + 3.23e3iT - 1.38e7T^{2}
67 1+5.93e3T+2.01e7T2 1 + 5.93e3T + 2.01e7T^{2}
71 1+8.19e3T+2.54e7T2 1 + 8.19e3T + 2.54e7T^{2}
73 1+6.90e3iT2.83e7T2 1 + 6.90e3iT - 2.83e7T^{2}
79 13.46e3iT3.89e7T2 1 - 3.46e3iT - 3.89e7T^{2}
83 1+7.71e3T+4.74e7T2 1 + 7.71e3T + 4.74e7T^{2}
89 12.86e3iT6.27e7T2 1 - 2.86e3iT - 6.27e7T^{2}
97 1+1.52e3T+8.85e7T2 1 + 1.52e3T + 8.85e7T^{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

−11.99597261157795625184799525260, −11.12193588216989091150538645877, −10.22776108075313501481106838785, −8.745663110581633329979635782230, −7.54168486046294545046189181485, −6.40642221657024911400631345893, −4.23149598696874856633382391412, −2.76015967441899488910601670980, −1.71974412646495322626749410634, −0.11974346179305530082886418936, 3.96095036413996811455380213674, 4.78907557384286616327298581306, 5.72756899710910252093142014399, 7.28324618371942487806551072564, 8.550604601254261833946188462114, 9.056648681887890549032867333872, 10.30527243236774313570614157502, 11.67499685370870310821050072302, 13.17569682843161815015939783839, 14.21320620194939977407638807099

Graph of the ZZ-function along the critical line