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Properties

Label 2-1638-1.1-c1-0-19
Degree 22
Conductor 16381638
Sign 11
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 3·11-s + 13-s + 14-s + 16-s − 3·17-s + 5·19-s + 3·20-s + 3·22-s − 3·23-s + 4·25-s + 26-s + 28-s − 3·29-s − 10·31-s + 32-s − 3·34-s + 3·35-s − 7·37-s + 5·38-s + 3·40-s + 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.670·20-s + 0.639·22-s − 0.625·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s − 0.557·29-s − 1.79·31-s + 0.176·32-s − 0.514·34-s + 0.507·35-s − 1.15·37-s + 0.811·38-s + 0.474·40-s + 0.937·41-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 1)(2,\ 1638,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.7278834473.727883447
L(12)L(\frac12) ≈\approx 3.7278834473.727883447
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−T 1 - T
3 1 1
7 1−T 1 - T
13 1−T 1 - T
good5 1−3T+pT2 1 - 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+T+pT2 1 + T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−9.324599393283520040395095380365, −8.841848699193986448317936408324, −7.53876669465829471955650703578, −6.83898930928988508530993461842, −5.85174835642101621758697196256, −5.52063493483957242179120678609, −4.39709509400492818170140602730, −3.48277098992130039606057870827, −2.22618758276738166969303174205, −1.46015232310843674903691285794, 1.46015232310843674903691285794, 2.22618758276738166969303174205, 3.48277098992130039606057870827, 4.39709509400492818170140602730, 5.52063493483957242179120678609, 5.85174835642101621758697196256, 6.83898930928988508530993461842, 7.53876669465829471955650703578, 8.841848699193986448317936408324, 9.324599393283520040395095380365

Graph of the ZZ-function along the critical line