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Properties

Label 2-234-13.12-c5-0-24
Degree 22
Conductor 234234
Sign −0.234+0.971i-0.234 + 0.971i
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 − 86.8i·5-s − 98.7i·7-s − 64i·8-s + 347.·10-s + 610. i·11-s + (592. + 143. i)13-s + 395.·14-s + 256·16-s + 1.14e3·17-s − 2.26e3i·19-s + 1.38e3i·20-s − 2.44e3·22-s − 433.·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.55i·5-s − 0.761i·7-s − 0.353i·8-s + 1.09·10-s + 1.52i·11-s + (0.971 + 0.234i)13-s + 0.538·14-s + 0.250·16-s + 0.963·17-s − 1.44i·19-s + 0.776i·20-s − 1.07·22-s − 0.170·23-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.234+0.971i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+5/2) L(s)=((−0.234+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.234+0.971i-0.234 + 0.971i
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), −0.234+0.971i)(2,\ 234,\ (\ :5/2),\ -0.234 + 0.971i)

Particular Values

L(3)L(3) ≈\approx 1.2619337621.261933762
L(12)L(\frac12) ≈\approx 1.2619337621.261933762
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+(−592.−143.i)T 1 + (-592. - 143. i)T
good5 1+86.8iT−3.12e3T2 1 + 86.8iT - 3.12e3T^{2}
7 1+98.7iT−1.68e4T2 1 + 98.7iT - 1.68e4T^{2}
11 1−610.iT−1.61e5T2 1 - 610. iT - 1.61e5T^{2}
17 1−1.14e3T+1.41e6T2 1 - 1.14e3T + 1.41e6T^{2}
19 1+2.26e3iT−2.47e6T2 1 + 2.26e3iT - 2.47e6T^{2}
23 1+433.T+6.43e6T2 1 + 433.T + 6.43e6T^{2}
29 1+7.66e3T+2.05e7T2 1 + 7.66e3T + 2.05e7T^{2}
31 1+7.36e3iT−2.86e7T2 1 + 7.36e3iT - 2.86e7T^{2}
37 1+1.05e4iT−6.93e7T2 1 + 1.05e4iT - 6.93e7T^{2}
41 1+3.69e3iT−1.15e8T2 1 + 3.69e3iT - 1.15e8T^{2}
43 1+6.06e3T+1.47e8T2 1 + 6.06e3T + 1.47e8T^{2}
47 1−8.74e3iT−2.29e8T2 1 - 8.74e3iT - 2.29e8T^{2}
53 1+3.47e4T+4.18e8T2 1 + 3.47e4T + 4.18e8T^{2}
59 1+1.19e4iT−7.14e8T2 1 + 1.19e4iT - 7.14e8T^{2}
61 1+4.54e4T+8.44e8T2 1 + 4.54e4T + 8.44e8T^{2}
67 1−4.56e4iT−1.35e9T2 1 - 4.56e4iT - 1.35e9T^{2}
71 1+2.38e4iT−1.80e9T2 1 + 2.38e4iT - 1.80e9T^{2}
73 1+3.77e4iT−2.07e9T2 1 + 3.77e4iT - 2.07e9T^{2}
79 1+3.53e4T+3.07e9T2 1 + 3.53e4T + 3.07e9T^{2}
83 1−3.14e4iT−3.93e9T2 1 - 3.14e4iT - 3.93e9T^{2}
89 1−5.90e4iT−5.58e9T2 1 - 5.90e4iT - 5.58e9T^{2}
97 1+4.96e3iT−8.58e9T2 1 + 4.96e3iT - 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

−11.02863556653891339517209650889, −9.612565490119138772806942403494, −9.145200425145832555467588288558, −7.918602933113753464778722624617, −7.19094174652952507656679588523, −5.78304801948556153484697870429, −4.73139053089582003258059214155, −3.99004334248554720880488743533, −1.59883081265575553430319927898, −0.37921355488310683196211631948, 1.54004811445409200498052538396, 3.13724758851073491568825200625, 3.47281876708367259058623711348, 5.62665225768673021523670576017, 6.26653386199517144480620083824, 7.80722662140288644923417936139, 8.688801069555330959361953351428, 9.962706446220888200175893222321, 10.76308597445775540388085946921, 11.39016413353933012732552657845

Graph of the ZZ-function along the critical line