Properties

Label 2-435-87.17-c1-0-3
Degree 22
Conductor 435435
Sign 0.680+0.733i0.680 + 0.733i
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  + (−1.48 − 1.48i)2-s + (−1.36 − 1.06i)3-s + 2.43i·4-s − 5-s + (0.442 + 3.61i)6-s − 1.31·7-s + (0.641 − 0.641i)8-s + (0.722 + 2.91i)9-s + (1.48 + 1.48i)10-s + (1.11 + 1.11i)11-s + (2.59 − 3.31i)12-s − 0.320i·13-s + (1.95 + 1.95i)14-s + (1.36 + 1.06i)15-s + 2.95·16-s + (0.771 + 0.771i)17-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + (−0.787 − 0.616i)3-s + 1.21i·4-s − 0.447·5-s + (0.180 + 1.47i)6-s − 0.495·7-s + (0.226 − 0.226i)8-s + (0.240 + 0.970i)9-s + (0.470 + 0.470i)10-s + (0.335 + 0.335i)11-s + (0.748 − 0.957i)12-s − 0.0889i·13-s + (0.521 + 0.521i)14-s + (0.352 + 0.275i)15-s + 0.737·16-s + (0.187 + 0.187i)17-s + ⋯

Functional equation

Λ(s)=(435s/2ΓC(s)L(s)=((0.680+0.733i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2ΓC(s+1/2)L(s)=((0.680+0.733i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    35293 \cdot 5 \cdot 29
Sign: 0.680+0.733i0.680 + 0.733i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(191,)\chi_{435} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.680+0.733i)(2,\ 435,\ (\ :1/2),\ 0.680 + 0.733i)

Particular Values

L(1)L(1) \approx 0.3999590.174497i0.399959 - 0.174497i
L(12)L(\frac12) \approx 0.3999590.174497i0.399959 - 0.174497i
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)
bad3 1+(1.36+1.06i)T 1 + (1.36 + 1.06i)T
5 1+T 1 + T
29 1+(0.1685.38i)T 1 + (-0.168 - 5.38i)T
good2 1+(1.48+1.48i)T+2iT2 1 + (1.48 + 1.48i)T + 2iT^{2}
7 1+1.31T+7T2 1 + 1.31T + 7T^{2}
11 1+(1.111.11i)T+11iT2 1 + (-1.11 - 1.11i)T + 11iT^{2}
13 1+0.320iT13T2 1 + 0.320iT - 13T^{2}
17 1+(0.7710.771i)T+17iT2 1 + (-0.771 - 0.771i)T + 17iT^{2}
19 1+(0.5620.562i)T19iT2 1 + (0.562 - 0.562i)T - 19iT^{2}
23 11.90iT23T2 1 - 1.90iT - 23T^{2}
31 1+(3.49+3.49i)T31iT2 1 + (-3.49 + 3.49i)T - 31iT^{2}
37 1+(0.2540.254i)T+37iT2 1 + (-0.254 - 0.254i)T + 37iT^{2}
41 1+(0.109+0.109i)T41iT2 1 + (-0.109 + 0.109i)T - 41iT^{2}
43 1+(3.91+3.91i)T43iT2 1 + (-3.91 + 3.91i)T - 43iT^{2}
47 1+(5.63+5.63i)T47iT2 1 + (-5.63 + 5.63i)T - 47iT^{2}
53 16.38iT53T2 1 - 6.38iT - 53T^{2}
59 1+1.03iT59T2 1 + 1.03iT - 59T^{2}
61 1+(4.99+4.99i)T61iT2 1 + (-4.99 + 4.99i)T - 61iT^{2}
67 111.6iT67T2 1 - 11.6iT - 67T^{2}
71 110.0T+71T2 1 - 10.0T + 71T^{2}
73 1+(9.529.52i)T+73iT2 1 + (-9.52 - 9.52i)T + 73iT^{2}
79 1+(2.052.05i)T79iT2 1 + (2.05 - 2.05i)T - 79iT^{2}
83 17.00iT83T2 1 - 7.00iT - 83T^{2}
89 1+(9.179.17i)T+89iT2 1 + (-9.17 - 9.17i)T + 89iT^{2}
97 1+(12.712.7i)T+97iT2 1 + (-12.7 - 12.7i)T + 97iT^{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

−11.00176034004970483847244536196, −10.28689695831280127253165491397, −9.436925119715321190673758099760, −8.397034731394763267544243934218, −7.52544723665090371543662049473, −6.53103120698840672275236048061, −5.33712713902563494376740281136, −3.78061632158098811023586836836, −2.34367457863809190828622754121, −0.965252506768556545714564581588, 0.59177154996541453191102467639, 3.42207333922996409433069980393, 4.70003408437216877628501510838, 5.99741176434406665835994340154, 6.55241697779050255746690284012, 7.56035191936343516769876321145, 8.585177676078175682085043566278, 9.401736569831760691257947625890, 10.09242796393090462916498535005, 11.01537137133939779218708317244

Graph of the ZZ-function along the critical line