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Properties

Label 4-440e2-1.1-c1e2-0-13
Degree 44
Conductor 193600193600
Sign −1-1
Analytic cond. 12.344112.3441
Root an. cond. 1.874411.87441
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 2·9-s + 8·19-s + 3·25-s + 8·35-s + 4·37-s + 20·43-s + 4·45-s − 2·49-s − 12·53-s + 8·63-s − 16·79-s − 5·81-s − 12·83-s − 12·89-s − 16·95-s + 4·97-s + 12·107-s − 12·113-s − 11·121-s − 4·125-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 2/3·9-s + 1.83·19-s + 3/5·25-s + 1.35·35-s + 0.657·37-s + 3.04·43-s + 0.596·45-s − 2/7·49-s − 1.64·53-s + 1.00·63-s − 1.80·79-s − 5/9·81-s − 1.31·83-s − 1.27·89-s − 1.64·95-s + 0.406·97-s + 1.16·107-s − 1.12·113-s − 121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(193600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(193600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 193600193600    =    26⋅52⋅1122^{6} \cdot 5^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 12.344112.3441
Root analytic conductor: 1.874411.87441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 193600, ( :1/2,1/2), −1)(4,\ 193600,\ (\ :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
5C1C_1 (1+T)2 ( 1 + T )^{2}
11C2C_2 1+pT2 1 + p T^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−8.740293324354028093320034492248, −8.597075432672291247955228431981, −7.71273110823499542846308181544, −7.48339952967373098458922175110, −7.12025897518162428207793885558, −6.27087624192875571051851265258, −6.09088770920212572077792597538, −5.46752609716119299096669229449, −4.83331995302090139893539895632, −4.12250433368686324236236368171, −3.61120105048083329633068803983, −2.90312545354050643736888796120, −2.76929890617261215013507568311, −1.17768193714929585417069498904, 0, 1.17768193714929585417069498904, 2.76929890617261215013507568311, 2.90312545354050643736888796120, 3.61120105048083329633068803983, 4.12250433368686324236236368171, 4.83331995302090139893539895632, 5.46752609716119299096669229449, 6.09088770920212572077792597538, 6.27087624192875571051851265258, 7.12025897518162428207793885558, 7.48339952967373098458922175110, 7.71273110823499542846308181544, 8.597075432672291247955228431981, 8.740293324354028093320034492248

Graph of the ZZ-function along the critical line