↑

Properties

Label 4-480e2-1.1-c1e2-0-51
Degree 44
Conductor 230400230400
Sign 11
Analytic cond. 14.690514.6905
Root an. cond. 1.957751.95775
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s + 9-s + 12·17-s + 25-s − 16·31-s − 12·41-s + 34·49-s + 8·63-s + 4·73-s − 16·79-s + 81-s + 36·89-s + 4·97-s + 8·103-s − 36·113-s + 96·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3.02·7-s + 1/3·9-s + 2.91·17-s + 1/5·25-s − 2.87·31-s − 1.87·41-s + 34/7·49-s + 1.00·63-s + 0.468·73-s − 1.80·79-s + 1/9·81-s + 3.81·89-s + 0.406·97-s + 0.788·103-s − 3.38·113-s + 8.80·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(230400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(230400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 230400230400    =    210⋅32⋅522^{10} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 14.690514.6905
Root analytic conductor: 1.957751.95775
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 230400, ( :1/2,1/2), 1)(4,\ 230400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8465409732.846540973
L(12)L(\frac12) ≈\approx 2.8465409732.846540973
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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + 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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 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 (1−18T+pT2)2 ( 1 - 18 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
show more
show less
   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

−9.082506062594783472831133921500, −8.307525687782695431948900879973, −7.965215938754459889106897012614, −7.71052407307967077644420749342, −7.38482725826507043429631860666, −6.76895238282715006434981339956, −5.76555679083117871916592673419, −5.48025531888955534300415720953, −4.95303891523898796353167575120, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −3.62987230449253956202768748221, −2.39041205999551187286748903960, −1.44399343799308961081305068602, −1.43294542786364492056174526795, 1.43294542786364492056174526795, 1.44399343799308961081305068602, 2.39041205999551187286748903960, 3.62987230449253956202768748221, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 4.95303891523898796353167575120, 5.48025531888955534300415720953, 5.76555679083117871916592673419, 6.76895238282715006434981339956, 7.38482725826507043429631860666, 7.71052407307967077644420749342, 7.965215938754459889106897012614, 8.307525687782695431948900879973, 9.082506062594783472831133921500

Graph of the ZZ-function along the critical line