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Properties

Label 2-234-13.12-c5-0-9
Degree 22
Conductor 234234
Sign −0.192−0.981i-0.192 - 0.981i
Analytic cond. 37.529837.5298
Root an. cond. 6.126156.12615
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 16·4-s − 51i·5-s + 105i·7-s − 64i·8-s + 204·10-s − 120i·11-s + (−598 + 117i)13-s − 420·14-s + 256·16-s + 1.10e3·17-s − 1.17e3i·19-s + 816i·20-s + 480·22-s − 1.05e3·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.912i·5-s + 0.809i·7-s − 0.353i·8-s + 0.645·10-s − 0.299i·11-s + (−0.981 + 0.192i)13-s − 0.572·14-s + 0.250·16-s + 0.923·17-s − 0.743i·19-s + 0.456i·20-s + 0.211·22-s − 0.413·23-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.192−0.981i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+5/2) L(s)=((−0.192−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.192−0.981i-0.192 - 0.981i
Analytic conductor: 37.529837.5298
Root analytic conductor: 6.126156.12615
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ234(181,⋅)\chi_{234} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :5/2), −0.192−0.981i)(2,\ 234,\ (\ :5/2),\ -0.192 - 0.981i)

Particular Values

L(3)L(3) ≈\approx 1.4770504401.477050440
L(12)L(\frac12) ≈\approx 1.4770504401.477050440
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
3 1 1
13 1+(598−117i)T 1 + (598 - 117i)T
good5 1+51iT−3.12e3T2 1 + 51iT - 3.12e3T^{2}
7 1−105iT−1.68e4T2 1 - 105iT - 1.68e4T^{2}
11 1+120iT−1.61e5T2 1 + 120iT - 1.61e5T^{2}
17 1−1.10e3T+1.41e6T2 1 - 1.10e3T + 1.41e6T^{2}
19 1+1.17e3iT−2.47e6T2 1 + 1.17e3iT - 2.47e6T^{2}
23 1+1.05e3T+6.43e6T2 1 + 1.05e3T + 6.43e6T^{2}
29 1−4.10e3T+2.05e7T2 1 - 4.10e3T + 2.05e7T^{2}
31 1−9.62e3iT−2.86e7T2 1 - 9.62e3iT - 2.86e7T^{2}
37 1−8.70e3iT−6.93e7T2 1 - 8.70e3iT - 6.93e7T^{2}
41 1−9.48e3iT−1.15e8T2 1 - 9.48e3iT - 1.15e8T^{2}
43 1−9.99e3T+1.47e8T2 1 - 9.99e3T + 1.47e8T^{2}
47 1−2.94e3iT−2.29e8T2 1 - 2.94e3iT - 2.29e8T^{2}
53 1−750T+4.18e8T2 1 - 750T + 4.18e8T^{2}
59 1−4.09e4iT−7.14e8T2 1 - 4.09e4iT - 7.14e8T^{2}
61 1+5.79e4T+8.44e8T2 1 + 5.79e4T + 8.44e8T^{2}
67 1−2.28e4iT−1.35e9T2 1 - 2.28e4iT - 1.35e9T^{2}
71 1+6.37e4iT−1.80e9T2 1 + 6.37e4iT - 1.80e9T^{2}
73 1−5.88e4iT−2.07e9T2 1 - 5.88e4iT - 2.07e9T^{2}
79 1−6.32e4T+3.07e9T2 1 - 6.32e4T + 3.07e9T^{2}
83 1+5.54e4iT−3.93e9T2 1 + 5.54e4iT - 3.93e9T^{2}
89 1−1.04e5iT−5.58e9T2 1 - 1.04e5iT - 5.58e9T^{2}
97 1−1.60e5iT−8.58e9T2 1 - 1.60e5iT - 8.58e9T^{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

−12.01564722989896368056223299065, −10.41389127526647543608719727596, −9.302176830956230348269183314698, −8.662558588919773944038673621331, −7.69034022700876328380984609525, −6.46717429904714964533298894295, −5.31584219160445818996791528294, −4.63755667528941151464541650240, −2.88290570232400841982222788082, −1.07623588320045590845175062375, 0.50840447317880579821769245286, 2.14130213880402589254586535111, 3.33124898745015316029784352709, 4.41321701233527808421709477405, 5.84144785520744976759359268610, 7.20402634972305475145049259457, 7.911105876696710420724617482656, 9.502633645548166488651127554080, 10.23679384610399907726125006126, 10.85941592912127044059202774695

Graph of the ZZ-function along the critical line