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Properties

Label 4-416e2-1.1-c1e2-0-30
Degree 44
Conductor 173056173056
Sign 11
Analytic cond. 11.034211.0342
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 3·7-s − 9-s − 4·11-s − 2·13-s + 3·15-s + 3·17-s − 12·19-s + 3·21-s + 25-s − 6·31-s + 4·33-s + 9·35-s − 7·37-s + 2·39-s + 6·41-s − 3·43-s + 3·45-s + 9·47-s − 3·49-s − 3·51-s − 18·53-s + 12·55-s + 12·57-s + 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1.13·7-s − 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.774·15-s + 0.727·17-s − 2.75·19-s + 0.654·21-s + 1/5·25-s − 1.07·31-s + 0.696·33-s + 1.52·35-s − 1.15·37-s + 0.320·39-s + 0.937·41-s − 0.457·43-s + 0.447·45-s + 1.31·47-s − 3/7·49-s − 0.420·51-s − 2.47·53-s + 1.61·55-s + 1.58·57-s + 1.56·59-s + 0.256·61-s + ⋯

Functional equation

Λ(s)=(173056s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(173056s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 173056173056    =    210⋅1322^{10} \cdot 13^{2}
Sign: 11
Analytic conductor: 11.034211.0342
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 173056, ( :1/2,1/2), 1)(4,\ 173056,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7D4D_{4} 1+3T+12T2+3pT3+p2T4 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17D4D_{4} 1−3T+32T2−3pT3+p2T4 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31D4D_{4} 1+6T+54T2+6pT3+p2T4 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+7T+48T2+7pT3+p2T4 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+74T2−6pT3+p2T4 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+3T−18T2+3pT3+p2T4 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−9T+76T2−9pT3+p2T4 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+18T+170T2+18pT3+p2T4 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4}
59C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
61D4D_{4} 1−2T−30T2−2pT3+p2T4 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4}
67C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
71D4D_{4} 1+9T+124T2+9pT3+p2T4 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
83D4D_{4} 1−10T+38T2−10pT3+p2T4 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.92935982090090269524877935962, −10.67565088589767443246463949915, −10.14961185105258290925313438877, −9.817551941800209256085620937533, −9.079843229105532359254548938110, −8.686642160719022342144432527557, −8.113845820974984917274798486075, −7.80661199171213693905190107623, −7.25185454608095247488917996700, −6.80989501401375884306401767831, −6.16947279325834065838914490210, −5.81669187073265291085738243765, −5.19032220429870854510795163877, −4.55268808714137228837513162197, −4.00928377650971953596887185380, −3.45306201677728978971783322968, −2.80712463094253039902674874006, −2.02601789412831845206354750598, 0, 0, 2.02601789412831845206354750598, 2.80712463094253039902674874006, 3.45306201677728978971783322968, 4.00928377650971953596887185380, 4.55268808714137228837513162197, 5.19032220429870854510795163877, 5.81669187073265291085738243765, 6.16947279325834065838914490210, 6.80989501401375884306401767831, 7.25185454608095247488917996700, 7.80661199171213693905190107623, 8.113845820974984917274798486075, 8.686642160719022342144432527557, 9.079843229105532359254548938110, 9.817551941800209256085620937533, 10.14961185105258290925313438877, 10.67565088589767443246463949915, 10.92935982090090269524877935962

Graph of the ZZ-function along the critical line