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Properties

Label 2-1332-148.147-c0-0-7
Degree 22
Conductor 13321332
Sign −1-1
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
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  − i·2-s − 4-s − 2i·5-s + i·8-s − 2·10-s + 16-s − 2i·17-s + 2i·20-s − 3·25-s + 2i·29-s − i·32-s − 2·34-s − 37-s + 2·40-s + 49-s + 3i·50-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 2i·5-s + i·8-s − 2·10-s + 16-s − 2i·17-s + 2i·20-s − 3·25-s + 2i·29-s − i·32-s − 2·34-s − 37-s + 2·40-s + 49-s + 3i·50-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: −1-1
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(739,⋅)\chi_{1332} (739, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), −1)(2,\ 1332,\ (\ :0),\ -1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.85188791110.8518879111
L(12)L(\frac12) ≈\approx 0.85188791110.8518879111
L(1)L(1) 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+iT 1 + iT
3 1 1
37 1+T 1 + T
good5 1+2iT−T2 1 + 2iT - T^{2}
7 1−T2 1 - T^{2}
11 1−T2 1 - T^{2}
13 1−T2 1 - T^{2}
17 1+2iT−T2 1 + 2iT - T^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1+T2 1 + T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−2T+T2 1 - 2T + T^{2}
79 1+T2 1 + T^{2}
83 1−T2 1 - T^{2}
89 1+2iT−T2 1 + 2iT - T^{2}
97 1−T2 1 - T^{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

−9.263316578553629191612984165716, −8.948198610174952362307836033265, −8.174508012512069959233592352296, −7.16664658589124262904547794499, −5.50116656458830640152099860584, −5.06240638219309506615955028832, −4.35031676942568073193058559590, −3.21418910234419495105039951092, −1.84593576095628764278001514242, −0.74639961767412490699413610569, 2.20670773972408657015567492181, 3.52071921081218638260797079824, 4.10210315070514634945543175068, 5.63725486358125138537236851318, 6.28692990200769658556860899977, 6.81329015461849506235384451688, 7.73669272123748992875920331166, 8.244531112739665959405529418217, 9.449464391846759011155041588538, 10.27430262381862409988665993183

Graph of the ZZ-function along the critical line