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Properties

Label 2-103428-1.1-c1-0-23
Degree 22
Conductor 103428103428
Sign −1-1
Analytic cond. 825.876825.876
Root an. cond. 28.738028.7380
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 17-s + 6·19-s + 8·23-s − 5·25-s − 10·29-s + 10·31-s + 2·37-s − 4·41-s + 12·43-s − 12·47-s − 3·49-s + 2·53-s + 10·61-s − 6·67-s − 4·71-s − 10·73-s − 8·79-s − 12·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.242·17-s + 1.37·19-s + 1.66·23-s − 25-s − 1.85·29-s + 1.79·31-s + 0.328·37-s − 0.624·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.274·53-s + 1.28·61-s − 0.733·67-s − 0.474·71-s − 1.17·73-s − 0.900·79-s − 1.27·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯

Functional equation

Λ(s)=(103428s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(103428s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 103428103428    =    22⋅32⋅132⋅172^{2} \cdot 3^{2} \cdot 13^{2} \cdot 17
Sign: −1-1
Analytic conductor: 825.876825.876
Root analytic conductor: 28.738028.7380
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 103428, ( :1/2), −1)(2,\ 103428,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
13 1 1
17 1+T 1 + T
good5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−6T+pT2 1 - 6 T + p 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

−14.15401619273066, −13.36249012802750, −13.07113670495462, −12.63766641979197, −11.67167189964776, −11.52712136387246, −11.32457601935092, −10.51027966738228, −10.01246055965879, −9.525856466187179, −9.026887349449478, −8.523652757911670, −7.906146444072465, −7.448346736254589, −7.125532963400858, −6.317567769079730, −5.819504401137548, −5.193137957836686, −4.830445276915432, −4.190240642975473, −3.532985917305665, −2.927501274580201, −2.324443818309210, −1.470579670710227, −1.053996848182529, 0, 1.053996848182529, 1.470579670710227, 2.324443818309210, 2.927501274580201, 3.532985917305665, 4.190240642975473, 4.830445276915432, 5.193137957836686, 5.819504401137548, 6.317567769079730, 7.125532963400858, 7.448346736254589, 7.906146444072465, 8.523652757911670, 9.026887349449478, 9.525856466187179, 10.01246055965879, 10.51027966738228, 11.32457601935092, 11.52712136387246, 11.67167189964776, 12.63766641979197, 13.07113670495462, 13.36249012802750, 14.15401619273066

Graph of the ZZ-function along the critical line