Properties

Label 2-435-435.434-c0-0-3
Degree 22
Conductor 435435
Sign 0.707+0.707i-0.707 + 0.707i
Analytic cond. 0.2170930.217093
Root an. cond. 0.4659320.465932
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯
L(s)  = 1  − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯

Functional equation

Λ(s)=(435s/2ΓC(s)L(s)=((0.707+0.707i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(435s/2ΓC(s)L(s)=((0.707+0.707i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    35293 \cdot 5 \cdot 29
Sign: 0.707+0.707i-0.707 + 0.707i
Analytic conductor: 0.2170930.217093
Root analytic conductor: 0.4659320.465932
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ435(434,)\chi_{435} (434, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :0), 0.707+0.707i)(2,\ 435,\ (\ :0),\ -0.707 + 0.707i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.65493467830.6549346783
L(12)L(\frac12) \approx 0.65493467830.6549346783
L(1)L(1) 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+(0.7070.707i)T 1 + (0.707 - 0.707i)T
5 1+iT 1 + iT
29 1iT 1 - iT
good2 1+1.41iTT2 1 + 1.41iT - T^{2}
7 1T2 1 - T^{2}
11 1+T2 1 + T^{2}
13 1T2 1 - T^{2}
17 1+1.41iTT2 1 + 1.41iT - T^{2}
19 1T2 1 - T^{2}
23 1+T2 1 + T^{2}
31 1T2 1 - T^{2}
37 11.41T+T2 1 - 1.41T + T^{2}
41 1+T2 1 + T^{2}
43 11.41T+T2 1 - 1.41T + T^{2}
47 11.41iTT2 1 - 1.41iT - T^{2}
53 1+T2 1 + T^{2}
59 12iTT2 1 - 2iT - T^{2}
61 1T2 1 - T^{2}
67 1T2 1 - T^{2}
71 1+2iTT2 1 + 2iT - T^{2}
73 1+1.41T+T2 1 + 1.41T + T^{2}
79 1T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1+1.41T+T2 1 + 1.41T + T^{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

−11.10925556937784685257595298515, −10.30107898932773866462678532612, −9.355172065686950677158247970746, −9.016485434287188897409415824302, −7.37611956487662343087107882024, −5.93876239627619688569353930404, −4.83169075143660255869303250892, −4.14429183287958096912215365830, −2.82987199154408230938686832926, −1.02405564282509788393045643942, 2.29200787907349334667327287820, 4.20323200289694540433527767246, 5.64776528050153626290529813582, 6.16702982690644972467450122874, 6.97099743013341101755526103891, 7.71690287397927697091285302851, 8.458656965702244809334768731076, 9.932589286928448295718445679514, 10.91973120262719832743795947517, 11.57612340855838048530728519379

Graph of the ZZ-function along the critical line