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Properties

Label 4-855e2-1.1-c1e2-0-7
Degree 44
Conductor 731025731025
Sign −1-1
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
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  − 4-s − 4·7-s + 6·13-s − 3·16-s + 6·19-s + 25-s + 4·28-s + 2·31-s − 8·37-s − 10·43-s + 2·49-s − 6·52-s + 4·61-s + 7·64-s − 14·67-s − 6·76-s + 4·79-s − 24·91-s + 2·97-s − 100-s + 28·103-s − 24·109-s + 12·112-s − 4·121-s − 2·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.51·7-s + 1.66·13-s − 3/4·16-s + 1.37·19-s + 1/5·25-s + 0.755·28-s + 0.359·31-s − 1.31·37-s − 1.52·43-s + 2/7·49-s − 0.832·52-s + 0.512·61-s + 7/8·64-s − 1.71·67-s − 0.688·76-s + 0.450·79-s − 2.51·91-s + 0.203·97-s − 0.0999·100-s + 2.75·103-s − 2.29·109-s + 1.13·112-s − 0.363·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 731025, ( :1/2,1/2), −1)(4,\ 731025,\ (\ :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)
bad3 1 1
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
19C2C_2 1−6T+pT2 1 - 6 T + p T^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−4T+pT2)(1−2T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )
17C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1−32T2+p2T4 1 - 32 T^{2} + p^{2} T^{4}
53C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
59C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
67C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
73C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−12T+pT2)(1+8T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+44T2+p2T4 1 + 44 T^{2} + p^{2} T^{4}
89C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−10T+pT2)(1+8T+pT2) ( 1 - 10 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

−8.255360423462715209168018903284, −7.53475293787070780905295708732, −7.17181001068942514933126862250, −6.55837165840372060726346851759, −6.38338293152944269940879376425, −5.93171982463535917019017484152, −5.22033215255672620850904885032, −4.98598676245323315788210901850, −4.17106231067447187066318367227, −3.70382491597435883966281116650, −3.24923691385426228788415144395, −2.92157462896746393067989153495, −1.87831161997820156325930422244, −1.06581621034428724597756516792, 0, 1.06581621034428724597756516792, 1.87831161997820156325930422244, 2.92157462896746393067989153495, 3.24923691385426228788415144395, 3.70382491597435883966281116650, 4.17106231067447187066318367227, 4.98598676245323315788210901850, 5.22033215255672620850904885032, 5.93171982463535917019017484152, 6.38338293152944269940879376425, 6.55837165840372060726346851759, 7.17181001068942514933126862250, 7.53475293787070780905295708732, 8.255360423462715209168018903284

Graph of the ZZ-function along the critical line