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Properties

Label 4-416e2-1.1-c1e2-0-18
Degree 44
Conductor 173056173056
Sign −1-1
Analytic cond. 11.034211.0342
Root an. cond. 1.822571.82257
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  − 5·9-s + 2·17-s − 4·23-s + 25-s + 4·29-s − 8·43-s + 3·49-s − 8·53-s − 4·61-s + 16·81-s − 8·101-s − 24·103-s − 24·107-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 10·153-s + 157-s + 163-s + 167-s − 13·169-s + 173-s + ⋯
L(s)  = 1  − 5/3·9-s + 0.485·17-s − 0.834·23-s + 1/5·25-s + 0.742·29-s − 1.21·43-s + 3/7·49-s − 1.09·53-s − 0.512·61-s + 16/9·81-s − 0.796·101-s − 2.36·103-s − 2.32·107-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.808·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 169-s + 0.0760·173-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: −1-1
Analytic conductor: 11.034211.0342
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
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
13C2C_2 1+pT2 1 + p T^{2}
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
5C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−5T+pT2)(1+3T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
31C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
37C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
41C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+T+pT2)(1+7T+pT2) ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} )
47C22C_2^2 1+29T2+p2T4 1 + 29 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
71C22C_2^2 1+101T2+p2T4 1 + 101 T^{2} + p^{2} T^{4}
73C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
79C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
83C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
89C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
97C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
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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

−8.890805373925362280572583914759, −8.381593766283564489885375705803, −8.096534916038210786810792941115, −7.70699316295590148500513561381, −6.88392073007506590632417046671, −6.46954662366170450227658639590, −5.98381614820652598219418737618, −5.40654780037387177486273649398, −5.08898276619219494971575841335, −4.28122546056690067414668236975, −3.63396173636179798667458470862, −2.95409560333340104380359722028, −2.51912929515045464280600200249, −1.44056631643362660336973086543, 0, 1.44056631643362660336973086543, 2.51912929515045464280600200249, 2.95409560333340104380359722028, 3.63396173636179798667458470862, 4.28122546056690067414668236975, 5.08898276619219494971575841335, 5.40654780037387177486273649398, 5.98381614820652598219418737618, 6.46954662366170450227658639590, 6.88392073007506590632417046671, 7.70699316295590148500513561381, 8.096534916038210786810792941115, 8.381593766283564489885375705803, 8.890805373925362280572583914759

Graph of the ZZ-function along the critical line