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Properties

Label 4-1331712-1.1-c1e2-0-14
Degree 44
Conductor 13317121331712
Sign −1-1
Analytic cond. 84.911184.9111
Root an. cond. 3.035573.03557
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  − 3·9-s + 4·17-s − 4·19-s − 4·25-s + 6·41-s + 8·43-s + 4·49-s − 16·59-s + 4·67-s − 2·73-s + 9·81-s + 16·83-s − 12·89-s − 10·97-s + 12·107-s − 34·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 9-s + 0.970·17-s − 0.917·19-s − 4/5·25-s + 0.937·41-s + 1.21·43-s + 4/7·49-s − 2.08·59-s + 0.488·67-s − 0.234·73-s + 81-s + 1.75·83-s − 1.27·89-s − 1.01·97-s + 1.16·107-s − 3.19·113-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(1331712s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1331712s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 13317121331712    =    29⋅32⋅1722^{9} \cdot 3^{2} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 84.911184.9111
Root analytic conductor: 3.035573.03557
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1331712, ( :1/2,1/2), −1)(4,\ 1331712,\ (\ :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
3C2C_2 1+pT2 1 + p T^{2}
17C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good5C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
7C22C_2^2 1−4T2+p2T4 1 - 4 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} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
29C22C_2^2 1+44T2+p2T4 1 + 44 T^{2} + p^{2} T^{4}
31C22C_2^2 1+52T2+p2T4 1 + 52 T^{2} + p^{2} T^{4}
37C22C_2^2 1+56T2+p2T4 1 + 56 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + 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}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C22C_2^2 1−88T2+p2T4 1 - 88 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C22C_2^2 1+104T2+p2T4 1 + 104 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+12T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} )
79C22C_2^2 1+36T2+p2T4 1 + 36 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+pT2)(1+12T+pT2) ( 1 + p T^{2} )( 1 + 12 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 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

−7.80969052675559674658849574077, −7.50348002922080914301099211794, −6.82096588531160295340893232549, −6.38728238601446323303977603504, −5.90815019937903916251268104929, −5.70093967452316249670673962656, −5.14748089057394383625273640312, −4.62245869586976571746118685581, −4.00219381662282336387092715427, −3.69948601322286334271165578008, −2.91917459370557589198931145184, −2.58767256442731917452415918296, −1.89376397280654085668022817676, −1.03807622769668816576661394988, 0, 1.03807622769668816576661394988, 1.89376397280654085668022817676, 2.58767256442731917452415918296, 2.91917459370557589198931145184, 3.69948601322286334271165578008, 4.00219381662282336387092715427, 4.62245869586976571746118685581, 5.14748089057394383625273640312, 5.70093967452316249670673962656, 5.90815019937903916251268104929, 6.38728238601446323303977603504, 6.82096588531160295340893232549, 7.50348002922080914301099211794, 7.80969052675559674658849574077

Graph of the ZZ-function along the critical line