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Properties

Label 4-345600-1.1-c1e2-0-26
Degree 44
Conductor 345600345600
Sign −1-1
Analytic cond. 22.035722.0357
Root an. cond. 2.166612.16661
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  − 3-s + 9-s + 4·17-s − 8·19-s − 25-s − 27-s + 8·41-s − 8·43-s − 2·49-s − 4·51-s + 8·57-s − 16·59-s + 8·67-s − 4·73-s + 75-s + 81-s − 8·83-s + 8·89-s − 28·97-s + 8·107-s − 20·113-s − 18·121-s − 8·123-s + 127-s + 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.970·17-s − 1.83·19-s − 1/5·25-s − 0.192·27-s + 1.24·41-s − 1.21·43-s − 2/7·49-s − 0.560·51-s + 1.05·57-s − 2.08·59-s + 0.977·67-s − 0.468·73-s + 0.115·75-s + 1/9·81-s − 0.878·83-s + 0.847·89-s − 2.84·97-s + 0.773·107-s − 1.88·113-s − 1.63·121-s − 0.721·123-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(345600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(345600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345600345600    =    29⋅33⋅522^{9} \cdot 3^{3} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 22.035722.0357
Root analytic conductor: 2.166612.16661
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 345600, ( :1/2,1/2), −1)(4,\ 345600,\ (\ :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
3C1C_1 1+T 1 + T
5C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
19C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
31C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
37C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
53C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1+6T+pT2)(1+10T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C22C_2^2 1+102T2+p2T4 1 + 102 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+10T+pT2)(1+18T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 18 T + p T^{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

−8.346291720121995717712093975596, −8.124319318961205390000648166586, −7.61948249871416466715769020252, −7.01807698922579834014571363934, −6.59751880432142696160633479249, −6.14527405832827378577925110928, −5.69478256156717491270165659232, −5.21106317421538041180495208875, −4.51900425270164728349791276127, −4.21343282699377366119345913002, −3.50903017322962956121673785600, −2.83664421549346698687301375081, −2.05129353552081938000857796892, −1.27841005985528659821354329133, 0, 1.27841005985528659821354329133, 2.05129353552081938000857796892, 2.83664421549346698687301375081, 3.50903017322962956121673785600, 4.21343282699377366119345913002, 4.51900425270164728349791276127, 5.21106317421538041180495208875, 5.69478256156717491270165659232, 6.14527405832827378577925110928, 6.59751880432142696160633479249, 7.01807698922579834014571363934, 7.61948249871416466715769020252, 8.124319318961205390000648166586, 8.346291720121995717712093975596

Graph of the ZZ-function along the critical line