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Properties

Label 4-975e2-1.1-c1e2-0-21
Degree 44
Conductor 950625950625
Sign −1-1
Analytic cond. 60.612660.6126
Root an. cond. 2.790232.79023
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  − 3-s − 4·4-s + 8·7-s − 2·9-s + 4·12-s − 2·13-s + 12·16-s − 8·19-s − 8·21-s + 5·27-s − 32·28-s − 8·31-s + 8·36-s − 4·37-s + 2·39-s + 14·43-s − 12·48-s + 34·49-s + 8·52-s + 8·57-s − 2·61-s − 16·63-s − 32·64-s − 28·67-s + 8·73-s + 32·76-s + 22·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 2·4-s + 3.02·7-s − 2/3·9-s + 1.15·12-s − 0.554·13-s + 3·16-s − 1.83·19-s − 1.74·21-s + 0.962·27-s − 6.04·28-s − 1.43·31-s + 4/3·36-s − 0.657·37-s + 0.320·39-s + 2.13·43-s − 1.73·48-s + 34/7·49-s + 1.10·52-s + 1.05·57-s − 0.256·61-s − 2.01·63-s − 4·64-s − 3.42·67-s + 0.936·73-s + 3.67·76-s + 2.47·79-s + ⋯

Functional equation

Λ(s)=(950625s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(950625s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 950625950625    =    32⋅54⋅1323^{2} \cdot 5^{4} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 60.612660.6126
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 950625, ( :1/2,1/2), −1)(4,\ 950625,\ (\ :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)
bad3C2C_2 1+T+pT2 1 + T + p T^{2}
5 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good2C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
29C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{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 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
73C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
79C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{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

−8.109622323430795384540902637182, −7.58470653085664427086815596244, −7.44313533967700149106082663018, −6.43995160346575362242193439929, −5.84441229372916213818129575211, −5.50324850311908281715674980863, −5.07836611320697081751291448911, −4.79815352058635420780275090547, −4.33348131529579942321198409567, −4.15253391774556813403352263543, −3.35745100328651024034839394504, −2.32784842593438550687981568602, −1.76397611793551290887183908518, −0.989438760515621806827446532171, 0, 0.989438760515621806827446532171, 1.76397611793551290887183908518, 2.32784842593438550687981568602, 3.35745100328651024034839394504, 4.15253391774556813403352263543, 4.33348131529579942321198409567, 4.79815352058635420780275090547, 5.07836611320697081751291448911, 5.50324850311908281715674980863, 5.84441229372916213818129575211, 6.43995160346575362242193439929, 7.44313533967700149106082663018, 7.58470653085664427086815596244, 8.109622323430795384540902637182

Graph of the ZZ-function along the critical line