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Properties

Label 4-234e2-1.1-c1e2-0-26
Degree 44
Conductor 5475654756
Sign −1-1
Analytic cond. 3.491293.49129
Root an. cond. 1.366931.36693
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·7-s − 2·13-s + 16-s − 12·19-s − 6·25-s − 4·28-s − 4·31-s + 12·37-s − 16·43-s − 2·49-s − 2·52-s + 20·61-s + 64-s − 4·67-s + 28·73-s − 12·76-s − 8·79-s + 8·91-s − 20·97-s − 6·100-s + 8·103-s + 20·109-s − 4·112-s − 6·121-s − 4·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s − 0.554·13-s + 1/4·16-s − 2.75·19-s − 6/5·25-s − 0.755·28-s − 0.718·31-s + 1.97·37-s − 2.43·43-s − 2/7·49-s − 0.277·52-s + 2.56·61-s + 1/8·64-s − 0.488·67-s + 3.27·73-s − 1.37·76-s − 0.900·79-s + 0.838·91-s − 2.03·97-s − 3/5·100-s + 0.788·103-s + 1.91·109-s − 0.377·112-s − 0.545·121-s − 0.359·124-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=(54756s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(54756s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5475654756    =    22⋅34⋅1322^{2} \cdot 3^{4} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 3.491293.49129
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 54756, ( :1/2,1/2), −1)(4,\ 54756,\ (\ :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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_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}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 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+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−9.773699703719295926726447008045, −9.511260567162705652960388658514, −8.695739660347841822203445826112, −8.241527102852606326983085030222, −7.76516545555292037952264969735, −6.93707785293490008685942511031, −6.49638165287752346862203973768, −6.33856513547641234727266817168, −5.59584593532556316161391417712, −4.80645914681781210957697738257, −3.99394475360161838020150844401, −3.53791924823882088462258917674, −2.57523735650192716532994366231, −2.01663812390586556648231292076, 0, 2.01663812390586556648231292076, 2.57523735650192716532994366231, 3.53791924823882088462258917674, 3.99394475360161838020150844401, 4.80645914681781210957697738257, 5.59584593532556316161391417712, 6.33856513547641234727266817168, 6.49638165287752346862203973768, 6.93707785293490008685942511031, 7.76516545555292037952264969735, 8.241527102852606326983085030222, 8.695739660347841822203445826112, 9.511260567162705652960388658514, 9.773699703719295926726447008045

Graph of the ZZ-function along the critical line