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Properties

Label 4-666e2-1.1-c1e2-0-10
Degree 44
Conductor 443556443556
Sign 11
Analytic cond. 28.281528.2815
Root an. cond. 2.306082.30608
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-s − 5-s + 8-s + 10-s − 4·11-s + 6·13-s − 16-s + 3·17-s − 2·19-s + 4·22-s + 12·23-s + 5·25-s − 6·26-s + 18·29-s + 20·31-s − 3·34-s − 37-s + 2·38-s − 40-s + 3·41-s − 16·43-s − 12·46-s − 4·47-s + 7·49-s − 5·50-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.852·22-s + 2.50·23-s + 25-s − 1.17·26-s + 3.34·29-s + 3.59·31-s − 0.514·34-s − 0.164·37-s + 0.324·38-s − 0.158·40-s + 0.468·41-s − 2.43·43-s − 1.76·46-s − 0.583·47-s + 49-s − 0.707·50-s − 0.824·53-s + 0.539·55-s + ⋯

Functional equation

Λ(s)=(443556s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(443556s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 443556443556    =    22⋅34⋅3722^{2} \cdot 3^{4} \cdot 37^{2}
Sign: 11
Analytic conductor: 28.281528.2815
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 443556, ( :1/2,1/2), 1)(4,\ 443556,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4238994521.423899452
L(12)L(\frac12) ≈\approx 1.4238994521.423899452
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)
bad2C2C_2 1+T+T2 1 + T + T^{2}
3 1 1
37C2C_2 1+T+pT2 1 + T + p T^{2}
good5C22C_2^2 1+T−4T2+pT3+p2T4 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}
7C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−3T−8T2−3pT3+p2T4 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C22C_2^2 1−3T−32T2−3pT3+p2T4 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−8T+5T2−8pT3+p2T4 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−5T−36T2−5pT3+p2T4 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−6T−31T2−6pT3+p2T4 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+6T−43T2+6pT3+p2T4 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−2T−79T2−2pT3+p2T4 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+13T+80T2+13pT3+p2T4 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4}
97C2C_2 (1−3T+pT2)2 ( 1 - 3 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

−10.56182993229973597377111061432, −10.43137175751561101728152793698, −9.842415721938944643242407657148, −9.620051478534108353908660449138, −8.605848493459808768427459796294, −8.517268320953162470790712985072, −8.250448953941239666876287998488, −8.142501523555363276404054290578, −7.05122773389641291060436914852, −6.90405443356493407183197926199, −6.44391746946068000561569181949, −5.92060136837988213742049818426, −5.03788140805041894714386836674, −4.83800202762840085182671713931, −4.43072838136009752251176800975, −3.48482059807355046542627915642, −2.82793741257533133459222477835, −2.76348563949483340092135235285, −1.05977880508536725700771349528, −1.01936612968966133430867180724, 1.01936612968966133430867180724, 1.05977880508536725700771349528, 2.76348563949483340092135235285, 2.82793741257533133459222477835, 3.48482059807355046542627915642, 4.43072838136009752251176800975, 4.83800202762840085182671713931, 5.03788140805041894714386836674, 5.92060136837988213742049818426, 6.44391746946068000561569181949, 6.90405443356493407183197926199, 7.05122773389641291060436914852, 8.142501523555363276404054290578, 8.250448953941239666876287998488, 8.517268320953162470790712985072, 8.605848493459808768427459796294, 9.620051478534108353908660449138, 9.842415721938944643242407657148, 10.43137175751561101728152793698, 10.56182993229973597377111061432

Graph of the ZZ-function along the critical line