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Properties

Label 4-170e2-1.1-c1e2-0-11
Degree 44
Conductor 2890028900
Sign 11
Analytic cond. 1.842681.84268
Root an. cond. 1.165091.16509
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s + 3·4-s − 8·6-s + 4·7-s − 4·8-s + 7·9-s − 4·11-s + 12·12-s + 2·13-s − 8·14-s + 5·16-s − 14·18-s + 2·19-s + 16·21-s + 8·22-s − 16·24-s + 25-s − 4·26-s + 4·27-s + 12·28-s − 4·31-s − 6·32-s − 16·33-s + 21·36-s − 12·37-s − 4·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s + 3/2·4-s − 3.26·6-s + 1.51·7-s − 1.41·8-s + 7/3·9-s − 1.20·11-s + 3.46·12-s + 0.554·13-s − 2.13·14-s + 5/4·16-s − 3.29·18-s + 0.458·19-s + 3.49·21-s + 1.70·22-s − 3.26·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 2.26·28-s − 0.718·31-s − 1.06·32-s − 2.78·33-s + 7/2·36-s − 1.97·37-s − 0.648·38-s + ⋯

Functional equation

Λ(s)=(28900s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(28900s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2890028900    =    22⋅52⋅1722^{2} \cdot 5^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 1.842681.84268
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 28900, ( :1/2,1/2), 1)(4,\ 28900,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4825904431.482590443
L(12)L(\frac12) ≈\approx 1.4825904431.482590443
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
17C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2×\timesC2C_2 (1−pT+pT2)(1−T+pT2) ( 1 - p T + p T^{2} )( 1 - T + p T^{2} )
7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2×\timesC2C_2 (1−5T+pT2)(1+3T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−3T+pT2)(1+T+pT2) ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
31C2C_2×\timesC2C_2 (1+T+pT2)(1+3T+pT2) ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+9T+pT2)(1+13T+pT2) ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} )
53C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
59C2C_2×\timesC2C_2 (1−15T+pT2)(1−3T+pT2) ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} )
61C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−14T+pT2)(1+2T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−9T+pT2)(1−3T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−11T+pT2)(1+3T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
83C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
89C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
97C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 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

−15.4802893459, −14.7042188078, −14.5856217610, −14.0366072852, −13.8085645982, −12.8899956802, −12.8337363923, −11.7739202135, −11.3100228916, −10.9484142878, −10.3636633894, −9.67317537122, −9.47750402875, −8.69848461419, −8.33817548631, −8.04714878525, −7.95836653673, −7.12227398289, −6.57312904075, −5.32145208242, −4.92015064778, −3.40344370945, −3.29460314858, −2.08881038737, −1.79579054759, 1.79579054759, 2.08881038737, 3.29460314858, 3.40344370945, 4.92015064778, 5.32145208242, 6.57312904075, 7.12227398289, 7.95836653673, 8.04714878525, 8.33817548631, 8.69848461419, 9.47750402875, 9.67317537122, 10.3636633894, 10.9484142878, 11.3100228916, 11.7739202135, 12.8337363923, 12.8899956802, 13.8085645982, 14.0366072852, 14.5856217610, 14.7042188078, 15.4802893459

Graph of the ZZ-function along the critical line