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Properties

Label 4-950e2-1.1-c3e2-0-3
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 3141.803141.80
Root an. cond. 7.486777.48677
Motivic weight 33
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  − 4·4-s + 50·9-s + 114·11-s + 16·16-s − 38·19-s + 300·29-s + 64·31-s − 200·36-s − 516·41-s − 456·44-s − 275·49-s + 660·59-s − 26·61-s − 64·64-s + 1.28e3·71-s + 152·76-s + 1.40e3·79-s + 1.77e3·81-s + 1.20e3·89-s + 5.70e3·99-s + 2.12e3·101-s − 2.92e3·109-s − 1.20e3·116-s + 7.08e3·121-s − 256·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.85·9-s + 3.12·11-s + 1/4·16-s − 0.458·19-s + 1.92·29-s + 0.370·31-s − 0.925·36-s − 1.96·41-s − 1.56·44-s − 0.801·49-s + 1.45·59-s − 0.0545·61-s − 1/8·64-s + 2.14·71-s + 0.229·76-s + 1.99·79-s + 2.42·81-s + 1.42·89-s + 5.78·99-s + 2.09·101-s − 2.56·109-s − 0.960·116-s + 5.32·121-s − 0.185·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 3141.803141.80
Root analytic conductor: 7.486777.48677
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :3/2,3/2), 1)(4,\ 902500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.5872553285.587255328
L(12)L(\frac12) ≈\approx 5.5872553285.587255328
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)
bad2C2C_2 1+p2T2 1 + p^{2} T^{2}
5 1 1
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1−50T2+p6T4 1 - 50 T^{2} + p^{6} T^{4}
7C22C_2^2 1+275T2+p6T4 1 + 275 T^{2} + p^{6} T^{4}
11C2C_2 (1−57T+p3T2)2 ( 1 - 57 T + p^{3} T^{2} )^{2}
13C2C_2 (1−6pT+p3T2)(1+6pT+p3T2) ( 1 - 6 p T + p^{3} T^{2} )( 1 + 6 p T + p^{3} T^{2} )
17C22C_2^2 1−5065T2+p6T4 1 - 5065 T^{2} + p^{6} T^{4}
23C22C_2^2 1−19150T2+p6T4 1 - 19150 T^{2} + p^{6} T^{4}
29C2C_2 (1−150T+p3T2)2 ( 1 - 150 T + p^{3} T^{2} )^{2}
31C2C_2 (1−32T+p3T2)2 ( 1 - 32 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−50230T2+p6T4 1 - 50230 T^{2} + p^{6} T^{4}
41C2C_2 (1+258T+p3T2)2 ( 1 + 258 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−154525T2+p6T4 1 - 154525 T^{2} + p^{6} T^{4}
47C22C_2^2 1+127595T2+p6T4 1 + 127595 T^{2} + p^{6} T^{4}
53C22C_2^2 1−111130T2+p6T4 1 - 111130 T^{2} + p^{6} T^{4}
59C2C_2 (1−330T+p3T2)2 ( 1 - 330 T + p^{3} T^{2} )^{2}
61C2C_2 (1+13T+p3T2)2 ( 1 + 13 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+131210T2+p6T4 1 + 131210 T^{2} + p^{6} T^{4}
71C2C_2 (1−642T+p3T2)2 ( 1 - 642 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−540865T2+p6T4 1 - 540865 T^{2} + p^{6} T^{4}
79C2C_2 (1−700T+p3T2)2 ( 1 - 700 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−1143430T2+p6T4 1 - 1143430 T^{2} + p^{6} T^{4}
89C2C_2 (1−600T+p3T2)2 ( 1 - 600 T + p^{3} T^{2} )^{2}
97C22C_2^2 1+202430T2+p6T4 1 + 202430 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

−9.644064037140311240561006068300, −9.546128984383800734137750149014, −9.161236716932795085658509517337, −8.673752622207650673948618893327, −8.194347193438798745591987146293, −7.957922448707580277618440977216, −6.97215459298714555403848428555, −6.91113085207857680330353429048, −6.48890752218509892622696000315, −6.36735309107665145319883590926, −5.48114803641410353242478146921, −4.67786630321155019324172541633, −4.65555582243016300614076263349, −4.07776084083849348591990500347, −3.60414352446249723225262212765, −3.37379483734888320502834456982, −2.13248276283789345642119631744, −1.70041086502442814865661357666, −1.02200080965530189226789347875, −0.78135078342163339690231969317, 0.78135078342163339690231969317, 1.02200080965530189226789347875, 1.70041086502442814865661357666, 2.13248276283789345642119631744, 3.37379483734888320502834456982, 3.60414352446249723225262212765, 4.07776084083849348591990500347, 4.65555582243016300614076263349, 4.67786630321155019324172541633, 5.48114803641410353242478146921, 6.36735309107665145319883590926, 6.48890752218509892622696000315, 6.91113085207857680330353429048, 6.97215459298714555403848428555, 7.957922448707580277618440977216, 8.194347193438798745591987146293, 8.673752622207650673948618893327, 9.161236716932795085658509517337, 9.546128984383800734137750149014, 9.644064037140311240561006068300

Graph of the ZZ-function along the critical line