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Properties

Label 4-1587e2-1.1-c1e2-0-5
Degree 44
Conductor 25185692518569
Sign 11
Analytic cond. 160.586160.586
Root an. cond. 3.559813.55981
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 − 2·3-s + 2·4-s − 2·5-s − 4·6-s + 4·7-s + 4·8-s + 3·9-s − 4·10-s + 6·11-s − 4·12-s + 2·13-s + 8·14-s + 4·15-s + 8·16-s + 10·17-s + 6·18-s + 4·19-s − 4·20-s − 8·21-s + 12·22-s − 8·24-s − 4·25-s + 4·26-s − 4·27-s + 8·28-s − 2·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.80·11-s − 1.15·12-s + 0.554·13-s + 2.13·14-s + 1.03·15-s + 2·16-s + 2.42·17-s + 1.41·18-s + 0.917·19-s − 0.894·20-s − 1.74·21-s + 2.55·22-s − 1.63·24-s − 4/5·25-s + 0.784·26-s − 0.769·27-s + 1.51·28-s − 0.371·29-s + ⋯

Functional equation

Λ(s)=(2518569s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2518569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2518569s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2518569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25185692518569    =    32⋅2343^{2} \cdot 23^{4}
Sign: 11
Analytic conductor: 160.586160.586
Root analytic conductor: 3.559813.55981
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2518569, ( :1/2,1/2), 1)(4,\ 2518569,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.9080543175.908054317
L(12)L(\frac12) ≈\approx 5.9080543175.908054317
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 (1+T)2 ( 1 + T )^{2}
23 1 1
good2C22C_2^2 1−pT+pT2−p2T3+p2T4 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4}
5D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
7D4D_{4} 1−4T+15T2−4pT3+p2T4 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−6T+28T2−6pT3+p2T4 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4}
13C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
17D4D_{4} 1−10T+56T2−10pT3+p2T4 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T−16T2+2pT3+p2T4 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37D4D_{4} 1−4T+51T2−4pT3+p2T4 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−12T+106T2−12pT3+p2T4 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−16T+147T2−16pT3+p2T4 1 - 16 T + 147 T^{2} - 16 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+76T2+6pT3+p2T4 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+62T2+4pT3+p2T4 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+2T+44T2+2pT3+p2T4 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+4T+99T2+4pT3+p2T4 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+24T+275T2+24pT3+p2T4 1 + 24 T + 275 T^{2} + 24 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+2T+116T2+2pT3+p2T4 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
79C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89D4D_{4} 1+6T+160T2+6pT3+p2T4 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+16T+210T2+16pT3+p2T4 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4}
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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

−9.710726678137462680201816035408, −9.273722835226276255482589970341, −8.888471249866983017775379926225, −8.034279992757575114953332093505, −7.65462685401747016378657310678, −7.61985594474241822079543889054, −7.46668596655168341563425077685, −6.62611056376319661147003609877, −6.05489698175622977188308942900, −5.88921889767383060529179561493, −5.43750904875789304335230630180, −5.10209092005732472593233544989, −4.40472903806115616935550056854, −4.39917985335952551983996414399, −3.73095764450750418493617157698, −3.69155295897364332914274447322, −2.89923121676509354618852588056, −1.73504804923049686111992196544, −1.30060884479117841379766125597, −1.03062855264119002155536407222, 1.03062855264119002155536407222, 1.30060884479117841379766125597, 1.73504804923049686111992196544, 2.89923121676509354618852588056, 3.69155295897364332914274447322, 3.73095764450750418493617157698, 4.39917985335952551983996414399, 4.40472903806115616935550056854, 5.10209092005732472593233544989, 5.43750904875789304335230630180, 5.88921889767383060529179561493, 6.05489698175622977188308942900, 6.62611056376319661147003609877, 7.46668596655168341563425077685, 7.61985594474241822079543889054, 7.65462685401747016378657310678, 8.034279992757575114953332093505, 8.888471249866983017775379926225, 9.273722835226276255482589970341, 9.710726678137462680201816035408

Graph of the ZZ-function along the critical line