Properties

Label 4-2850e2-1.1-c1e2-0-26
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 16-s − 2·19-s − 4·29-s − 4·31-s + 36-s − 24·41-s + 10·49-s − 12·59-s − 28·61-s − 64-s − 16·71-s + 2·76-s − 28·79-s + 81-s − 12·101-s + 24·109-s + 4·116-s − 22·121-s + 4·124-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 − 0.458·19-s − 0.742·29-s − 0.718·31-s + 1/6·36-s − 3.74·41-s + 10/7·49-s − 1.56·59-s − 3.58·61-s − 1/8·64-s − 1.89·71-s + 0.229·76-s − 3.15·79-s + 1/9·81-s − 1.19·101-s + 2.29·109-s + 0.371·116-s − 2·121-s + 0.359·124-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.609339953980742362499775860897, −8.331948716130057129283228605701, −7.893978685593130274276193065219, −7.42098781894316868418425688240, −7.10393188082123719791975277481, −6.85055130282602910773318627121, −6.07061188467006871523874868514, −5.97409249623160968496874865630, −5.59968821722907811729320573093, −5.01398874952407618454535122645, −4.67128761218873755355165959362, −4.35522490680741877412835288664, −3.79503386535108056159683250107, −3.21378981627712944750297954409, −3.11585862337281089125486532781, −2.36376435035586883623279974902, −1.53079113486004928877448023407, −1.51957875104785536736135194568, 0, 0, 1.51957875104785536736135194568, 1.53079113486004928877448023407, 2.36376435035586883623279974902, 3.11585862337281089125486532781, 3.21378981627712944750297954409, 3.79503386535108056159683250107, 4.35522490680741877412835288664, 4.67128761218873755355165959362, 5.01398874952407618454535122645, 5.59968821722907811729320573093, 5.97409249623160968496874865630, 6.07061188467006871523874868514, 6.85055130282602910773318627121, 7.10393188082123719791975277481, 7.42098781894316868418425688240, 7.893978685593130274276193065219, 8.331948716130057129283228605701, 8.609339953980742362499775860897

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