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Properties

Label 2-2352-4.3-c2-0-66
Degree 22
Conductor 23522352
Sign −0.866+0.5i-0.866 + 0.5i
Analytic cond. 64.087364.0873
Root an. cond. 8.005458.00545
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + 0.417·5-s − 2.99·9-s + 18.4i·11-s + 1.16·13-s − 0.723i·15-s − 0.417·17-s − 21.1i·19-s + 16.9i·23-s − 24.8·25-s + 5.19i·27-s + 4.33·29-s − 20.7i·31-s + 31.9·33-s − 61.1·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.0834·5-s − 0.333·9-s + 1.67i·11-s + 0.0896·13-s − 0.0482i·15-s − 0.0245·17-s − 1.11i·19-s + 0.738i·23-s − 0.993·25-s + 0.192i·27-s + 0.149·29-s − 0.670i·31-s + 0.967·33-s − 1.65·37-s + ⋯

Functional equation

Λ(s)=(2352s/2 ΓC(s) L(s)=((−0.866+0.5i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2352s/2 ΓC(s+1) L(s)=((−0.866+0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23522352    =    24⋅3⋅722^{4} \cdot 3 \cdot 7^{2}
Sign: −0.866+0.5i-0.866 + 0.5i
Analytic conductor: 64.087364.0873
Root analytic conductor: 8.005458.00545
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2352(1471,⋅)\chi_{2352} (1471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2352, ( :1), −0.866+0.5i)(2,\ 2352,\ (\ :1),\ -0.866 + 0.5i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.69390954260.6939095426
L(12)L(\frac12) ≈\approx 0.69390954260.6939095426
L(2)L(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 1
3 1+1.73iT 1 + 1.73iT
7 1 1
good5 1−0.417T+25T2 1 - 0.417T + 25T^{2}
11 1−18.4iT−121T2 1 - 18.4iT - 121T^{2}
13 1−1.16T+169T2 1 - 1.16T + 169T^{2}
17 1+0.417T+289T2 1 + 0.417T + 289T^{2}
19 1+21.1iT−361T2 1 + 21.1iT - 361T^{2}
23 1−16.9iT−529T2 1 - 16.9iT - 529T^{2}
29 1−4.33T+841T2 1 - 4.33T + 841T^{2}
31 1+20.7iT−961T2 1 + 20.7iT - 961T^{2}
37 1+61.1T+1.36e3T2 1 + 61.1T + 1.36e3T^{2}
41 1−9.07T+1.68e3T2 1 - 9.07T + 1.68e3T^{2}
43 1−7.30iT−1.84e3T2 1 - 7.30iT - 1.84e3T^{2}
47 1+19.3iT−2.20e3T2 1 + 19.3iT - 2.20e3T^{2}
53 1−92.1T+2.80e3T2 1 - 92.1T + 2.80e3T^{2}
59 1+99.2iT−3.48e3T2 1 + 99.2iT - 3.48e3T^{2}
61 1+78.6T+3.72e3T2 1 + 78.6T + 3.72e3T^{2}
67 1+77.6iT−4.48e3T2 1 + 77.6iT - 4.48e3T^{2}
71 1+43.9iT−5.04e3T2 1 + 43.9iT - 5.04e3T^{2}
73 1−53.8T+5.32e3T2 1 - 53.8T + 5.32e3T^{2}
79 1+74.7iT−6.24e3T2 1 + 74.7iT - 6.24e3T^{2}
83 1+32.5iT−6.88e3T2 1 + 32.5iT - 6.88e3T^{2}
89 1+81.9T+7.92e3T2 1 + 81.9T + 7.92e3T^{2}
97 1−30.1T+9.40e3T2 1 - 30.1T + 9.40e3T^{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

−8.427251776165363770904698036016, −7.49824616549153332317428287264, −7.10781182033392105934296085316, −6.27441119755722482957328493416, −5.29700892015897448170576304935, −4.56134398525228240916183845795, −3.52987317870867121183152822677, −2.31751462900379023057927650987, −1.65205393825054313573273720860, −0.16982063156988279293594728730, 1.17436538884681563788121685552, 2.57207772985668609739763713094, 3.52548102150223635396273207390, 4.13559220699278787562756502109, 5.37841282198820942285490473684, 5.82282639174542888148321607909, 6.68037338475128260962351005599, 7.75461871322066235253043288860, 8.578366384180062155618568931136, 8.895752616844363872678636431190

Graph of the ZZ-function along the critical line