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Properties

Label 4-416e2-1.1-c1e2-0-17
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  − 2·5-s − 9-s + 2·13-s − 2·17-s + 8·19-s − 8·23-s − 3·25-s − 10·37-s + 2·45-s − 5·49-s + 8·59-s − 4·65-s − 8·79-s − 8·81-s + 8·83-s + 4·85-s − 16·95-s + 8·103-s − 18·109-s + 4·113-s + 16·115-s − 2·117-s − 22·121-s + 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s + 0.554·13-s − 0.485·17-s + 1.83·19-s − 1.66·23-s − 3/5·25-s − 1.64·37-s + 0.298·45-s − 5/7·49-s + 1.04·59-s − 0.496·65-s − 0.900·79-s − 8/9·81-s + 0.878·83-s + 0.433·85-s − 1.64·95-s + 0.788·103-s − 1.72·109-s + 0.376·113-s + 1.49·115-s − 0.184·117-s − 2·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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 1−2T+pT2 1 - 2 T + p T^{2}
good3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C2C_2×\timesC2C_2 (1−T+pT2)(1+3T+pT2) ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} )
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2×\timesC2C_2 (1−3T+pT2)(1+5T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
23C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1+3T+pT2)(1+7T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} )
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1−63T2+p2T4 1 - 63 T^{2} + p^{2} T^{4}
47C22C_2^2 1+5T2+p2T4 1 + 5 T^{2} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C22C_2^2 1−11T2+p2T4 1 - 11 T^{2} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C22C_2^2 1+62T2+p2T4 1 + 62 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

−8.875695304099684896281182707766, −8.420945047513015779175356935662, −7.935387512355577643087283909151, −7.65104784567127020197799894934, −7.06455621115253245281313072600, −6.56480351600023000949595423956, −5.91753095980449469480233466188, −5.47293856182412718696639704946, −4.93230086189046741029681749520, −4.13991592642427677936275988561, −3.71352235547140785875774871076, −3.24216290472028332104415975278, −2.33989940612046940156518447462, −1.40740953620497473077259660886, 0, 1.40740953620497473077259660886, 2.33989940612046940156518447462, 3.24216290472028332104415975278, 3.71352235547140785875774871076, 4.13991592642427677936275988561, 4.93230086189046741029681749520, 5.47293856182412718696639704946, 5.91753095980449469480233466188, 6.56480351600023000949595423956, 7.06455621115253245281313072600, 7.65104784567127020197799894934, 7.935387512355577643087283909151, 8.420945047513015779175356935662, 8.875695304099684896281182707766

Graph of the ZZ-function along the critical line