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Properties

Label 4-525e2-1.1-c1e2-0-3
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 17.574017.5740
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 8·8-s + 9-s − 8·11-s − 7·16-s + 2·18-s − 16·22-s − 4·29-s + 14·32-s − 36-s + 20·37-s − 8·43-s + 8·44-s − 7·49-s + 20·53-s − 8·58-s + 35·64-s − 24·67-s − 16·71-s − 8·72-s + 40·74-s + 81-s − 16·86-s + 64·88-s − 14·98-s − 8·99-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s + 1/3·9-s − 2.41·11-s − 7/4·16-s + 0.471·18-s − 3.41·22-s − 0.742·29-s + 2.47·32-s − 1/6·36-s + 3.28·37-s − 1.21·43-s + 1.20·44-s − 49-s + 2.74·53-s − 1.05·58-s + 35/8·64-s − 2.93·67-s − 1.89·71-s − 0.942·72-s + 4.64·74-s + 1/9·81-s − 1.72·86-s + 6.82·88-s − 1.41·98-s − 0.804·99-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 17.574017.5740
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :1/2,1/2), 1)(4,\ 275625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3520253111.352025311
L(12)L(\frac12) ≈\approx 1.3520253111.352025311
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)
bad3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5 1 1
7C2C_2 1+pT2 1 + p T^{2}
good2C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 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 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 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+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−8.843956595114680827452383410006, −8.461212909606671780489016897710, −7.78056789512660315702386216522, −7.68003844385765985083943135134, −6.92482379445403167319259433221, −6.00448844798074713207333092038, −5.86497653842965752491328535183, −5.44640444640032659769086760535, −4.79896242020124990959414772446, −4.52160444884874669934496061646, −4.11644977935212568952932430273, −3.13361812930802426100944548657, −3.01549784140686353642765162463, −2.20286953987450620546650146772, −0.53965130087831198857502799684, 0.53965130087831198857502799684, 2.20286953987450620546650146772, 3.01549784140686353642765162463, 3.13361812930802426100944548657, 4.11644977935212568952932430273, 4.52160444884874669934496061646, 4.79896242020124990959414772446, 5.44640444640032659769086760535, 5.86497653842965752491328535183, 6.00448844798074713207333092038, 6.92482379445403167319259433221, 7.68003844385765985083943135134, 7.78056789512660315702386216522, 8.461212909606671780489016897710, 8.843956595114680827452383410006

Graph of the ZZ-function along the critical line