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Properties

Label 4-15e4-1.1-c3e2-0-10
Degree 44
Conductor 5062550625
Sign 11
Analytic cond. 176.237176.237
Root an. cond. 3.643543.64354
Motivic weight 33
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  + 15·4-s + 86·11-s + 161·16-s + 70·19-s + 320·29-s + 84·31-s + 406·41-s + 1.29e3·44-s + 650·49-s − 560·59-s − 1.03e3·61-s + 1.45e3·64-s − 824·71-s + 1.05e3·76-s − 1.02e3·79-s − 1.89e3·89-s − 2.60e3·101-s − 2.14e3·109-s + 4.80e3·116-s + 2.88e3·121-s + 1.26e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 15/8·4-s + 2.35·11-s + 2.51·16-s + 0.845·19-s + 2.04·29-s + 0.486·31-s + 1.54·41-s + 4.41·44-s + 1.89·49-s − 1.23·59-s − 2.17·61-s + 2.84·64-s − 1.37·71-s + 1.58·76-s − 1.45·79-s − 2.25·89-s − 2.56·101-s − 1.88·109-s + 3.84·116-s + 2.16·121-s + 0.912·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

Λ(s)=(50625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(50625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5062550625    =    34⋅543^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 176.237176.237
Root analytic conductor: 3.643543.64354
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 50625, ( :3/2,3/2), 1)(4,\ 50625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.6712733215.671273321
L(12)L(\frac12) ≈\approx 5.6712733215.671273321
L(52)L(\frac{5}{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)
bad3 1 1
5 1 1
good2C22C_2^2 1−15T2+p6T4 1 - 15 T^{2} + p^{6} T^{4}
7C22C_2^2 1−650T2+p6T4 1 - 650 T^{2} + p^{6} T^{4}
11C2C_2 (1−43T+p3T2)2 ( 1 - 43 T + p^{3} T^{2} )^{2}
13C22C_2^2 1−3610T2+p6T4 1 - 3610 T^{2} + p^{6} T^{4}
17C22C_2^2 1−1545T2+p6T4 1 - 1545 T^{2} + p^{6} T^{4}
19C2C_2 (1−35T+p3T2)2 ( 1 - 35 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+1910T2+p6T4 1 + 1910 T^{2} + p^{6} T^{4}
29C2C_2 (1−160T+p3T2)2 ( 1 - 160 T + p^{3} T^{2} )^{2}
31C2C_2 (1−42T+p3T2)2 ( 1 - 42 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−2710T2+p6T4 1 - 2710 T^{2} + p^{6} T^{4}
41C2C_2 (1−203T+p3T2)2 ( 1 - 203 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−150550T2+p6T4 1 - 150550 T^{2} + p^{6} T^{4}
47C22C_2^2 1−169230T2+p6T4 1 - 169230 T^{2} + p^{6} T^{4}
53C22C_2^2 1−291030T2+p6T4 1 - 291030 T^{2} + p^{6} T^{4}
59C2C_2 (1+280T+p3T2)2 ( 1 + 280 T + p^{3} T^{2} )^{2}
61C2C_2 (1+518T+p3T2)2 ( 1 + 518 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−581645T2+p6T4 1 - 581645 T^{2} + p^{6} T^{4}
71C2C_2 (1+412T+p3T2)2 ( 1 + 412 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−195865T2+p6T4 1 - 195865 T^{2} + p^{6} T^{4}
79C2C_2 (1+510T+p3T2)2 ( 1 + 510 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−539845T2+p6T4 1 - 539845 T^{2} + p^{6} T^{4}
89C2C_2 (1+945T+p3T2)2 ( 1 + 945 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−272830T2+p6T4 1 - 272830 T^{2} + p^{6} 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

−11.91024676504293148752800662682, −11.78095794529568528931988329931, −11.01358803702866093824695124426, −10.84989625072399154783647284807, −10.13650849674616674161009520178, −9.737666004305842109144221759153, −9.045275310979518361918703915143, −8.703886115521945247123125525885, −7.81631347999061760717321019399, −7.49758683097558234352435081383, −6.73218584991688115356403756762, −6.64904153775204110324849651449, −6.01895959926985832442616390006, −5.57983661243711358920732504523, −4.42126816620113659008484445807, −3.97636953211741518269571920036, −2.96998460118144349797316446203, −2.67550728019049928456824006338, −1.33272796765119163125405994647, −1.26783699447599961263571658716, 1.26783699447599961263571658716, 1.33272796765119163125405994647, 2.67550728019049928456824006338, 2.96998460118144349797316446203, 3.97636953211741518269571920036, 4.42126816620113659008484445807, 5.57983661243711358920732504523, 6.01895959926985832442616390006, 6.64904153775204110324849651449, 6.73218584991688115356403756762, 7.49758683097558234352435081383, 7.81631347999061760717321019399, 8.703886115521945247123125525885, 9.045275310979518361918703915143, 9.737666004305842109144221759153, 10.13650849674616674161009520178, 10.84989625072399154783647284807, 11.01358803702866093824695124426, 11.78095794529568528931988329931, 11.91024676504293148752800662682

Graph of the ZZ-function along the critical line