Properties

Label 4-1950e2-1.1-c1e2-0-1
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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  − 4-s − 9-s + 16-s − 8·19-s − 4·29-s − 16·31-s + 36-s − 12·41-s − 2·49-s − 20·61-s − 64-s − 32·71-s + 8·76-s + 16·79-s + 81-s + 28·89-s + 20·101-s − 28·109-s + 4·116-s − 22·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s − 1.83·19-s − 0.742·29-s − 2.87·31-s + 1/6·36-s − 1.87·41-s − 2/7·49-s − 2.56·61-s − 1/8·64-s − 3.79·71-s + 0.917·76-s + 1.80·79-s + 1/9·81-s + 2.96·89-s + 1.99·101-s − 2.68·109-s + 0.371·116-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1239765230\)
\(L(\frac12)\) \(\approx\) \(0.1239765230\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−9.520858046602122531956080282807, −8.912015836345967628129578832244, −8.769533074960395694331806283167, −8.293329139724881230151792418400, −7.77352877070571404646849925393, −7.49539115064719562021048168421, −7.13029841772736272071075069908, −6.46997782655535294855748051466, −6.29255337845808473098709614573, −5.69297987844155100951583072706, −5.48864365662071952858843922487, −4.72090186322783236501216952555, −4.68137354438211303368773936535, −3.98975436727305625103784005231, −3.46561640624047238436494549463, −3.31043715045951737008139054179, −2.41616805805092449875559807971, −1.88698152057395783200614594755, −1.48269543757084211639763743218, −0.12373399886804670219202394827, 0.12373399886804670219202394827, 1.48269543757084211639763743218, 1.88698152057395783200614594755, 2.41616805805092449875559807971, 3.31043715045951737008139054179, 3.46561640624047238436494549463, 3.98975436727305625103784005231, 4.68137354438211303368773936535, 4.72090186322783236501216952555, 5.48864365662071952858843922487, 5.69297987844155100951583072706, 6.29255337845808473098709614573, 6.46997782655535294855748051466, 7.13029841772736272071075069908, 7.49539115064719562021048168421, 7.77352877070571404646849925393, 8.293329139724881230151792418400, 8.769533074960395694331806283167, 8.912015836345967628129578832244, 9.520858046602122531956080282807

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