Properties

Label 4-980e2-1.1-c1e2-0-4
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 3·9-s − 6·11-s − 4·13-s − 15-s − 6·17-s + 8·19-s − 3·23-s + 8·27-s + 6·29-s + 2·31-s − 6·33-s − 8·37-s − 4·39-s + 6·41-s + 10·43-s − 3·45-s − 6·51-s − 12·53-s + 6·55-s + 8·57-s − 61-s + 4·65-s + 7·67-s − 3·69-s − 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 9-s − 1.80·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.625·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.447·45-s − 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s − 0.128·61-s + 0.496·65-s + 0.855·67-s − 0.361·69-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.624585999\)
\(L(\frac12)\) \(\approx\) \(1.624585999\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.22665326279681432802530989079, −9.750863008896782505080634156278, −9.364108686856853557880373933773, −9.113886758065741616957605798370, −8.281360903314058443916661374250, −8.188250272607580126965554398450, −7.74296214996388938098857911198, −7.27933364803808815766743741765, −7.03918237554094277966136532677, −6.57048773898033294378105196502, −5.83499383544825359569944306323, −5.34470118528749854473689820698, −4.80882369970073147010761300571, −4.57592037970624273610771017081, −4.06483709625315044635575586511, −3.23958708758297670858193642353, −2.77635348021519009364105900721, −2.46747832701378377104507718087, −1.64973143659609205796173223313, −0.56245545339233962096906289444, 0.56245545339233962096906289444, 1.64973143659609205796173223313, 2.46747832701378377104507718087, 2.77635348021519009364105900721, 3.23958708758297670858193642353, 4.06483709625315044635575586511, 4.57592037970624273610771017081, 4.80882369970073147010761300571, 5.34470118528749854473689820698, 5.83499383544825359569944306323, 6.57048773898033294378105196502, 7.03918237554094277966136532677, 7.27933364803808815766743741765, 7.74296214996388938098857911198, 8.188250272607580126965554398450, 8.281360903314058443916661374250, 9.113886758065741616957605798370, 9.364108686856853557880373933773, 9.750863008896782505080634156278, 10.22665326279681432802530989079

Graph of the $Z$-function along the critical line