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Properties

Label 4-416e2-1.1-c1e2-0-11
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 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·5-s − 9-s − 2·13-s − 6·17-s + 17·25-s + 20·29-s + 6·37-s − 6·45-s − 9·49-s + 8·53-s − 12·65-s + 28·73-s − 8·81-s − 36·85-s − 20·89-s − 4·97-s − 24·101-s + 22·109-s − 28·113-s + 2·117-s − 2·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 120·145-s + ⋯
L(s)  = 1  + 2.68·5-s − 1/3·9-s − 0.554·13-s − 1.45·17-s + 17/5·25-s + 3.71·29-s + 0.986·37-s − 0.894·45-s − 9/7·49-s + 1.09·53-s − 1.48·65-s + 3.27·73-s − 8/9·81-s − 3.90·85-s − 2.11·89-s − 0.406·97-s − 2.38·101-s + 2.10·109-s − 2.63·113-s + 0.184·117-s − 0.181·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.96·145-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: 00
Selberg data: (4, 173056, ( :1/2,1/2), 1)(4,\ 173056,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6500692372.650069237
L(12)L(\frac12) ≈\approx 2.6500692372.650069237
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}
good3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
7C22C_2^2 1+9T2+p2T4 1 + 9 T^{2} + p^{2} T^{4}
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
23C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1+41T2+p2T4 1 + 41 T^{2} + p^{2} T^{4}
47C22C_2^2 1+89T2+p2T4 1 + 89 T^{2} + p^{2} T^{4}
53C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
59C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
71C22C_2^2 1+97T2+p2T4 1 + 97 T^{2} + p^{2} T^{4}
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
83C22C_2^2 1−154T2+p2T4 1 - 154 T^{2} + p^{2} T^{4}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 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

−11.27110648180870015477247594486, −10.90615349797839564088435717397, −10.26361615459714924439044485613, −10.15917061080704108074497661375, −9.583026452911920925962200783250, −9.392098843380701832603579291700, −8.826516368022426000035757705537, −8.314822912965428696565169918059, −7.964788732557451643835501218434, −6.81927019276186980443433613696, −6.53859565101617360290440907337, −6.49409899418294038372661757501, −5.59339610543323639627859383173, −5.41266641732841925902519708962, −4.68488467222293078851897360358, −4.28579371422244664849394136141, −2.95529952262336145499629650859, −2.54111305152544911453321665086, −2.09393659444886036309468602636, −1.15759207088729262449011784997, 1.15759207088729262449011784997, 2.09393659444886036309468602636, 2.54111305152544911453321665086, 2.95529952262336145499629650859, 4.28579371422244664849394136141, 4.68488467222293078851897360358, 5.41266641732841925902519708962, 5.59339610543323639627859383173, 6.49409899418294038372661757501, 6.53859565101617360290440907337, 6.81927019276186980443433613696, 7.964788732557451643835501218434, 8.314822912965428696565169918059, 8.826516368022426000035757705537, 9.392098843380701832603579291700, 9.583026452911920925962200783250, 10.15917061080704108074497661375, 10.26361615459714924439044485613, 10.90615349797839564088435717397, 11.27110648180870015477247594486

Graph of the ZZ-function along the critical line