Properties

Label 4-2850e2-1.1-c1e2-0-5
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 4·11-s + 16-s − 2·19-s + 36-s − 16·41-s + 4·44-s + 10·49-s + 16·59-s + 4·61-s − 64-s + 16·71-s + 2·76-s + 16·79-s + 81-s − 32·89-s + 4·99-s + 4·101-s − 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.20·11-s + 1/4·16-s − 0.458·19-s + 1/6·36-s − 2.49·41-s + 0.603·44-s + 10/7·49-s + 2.08·59-s + 0.512·61-s − 1/8·64-s + 1.89·71-s + 0.229·76-s + 1.80·79-s + 1/9·81-s − 3.39·89-s + 0.402·99-s + 0.398·101-s − 0.383·109-s − 0.909·121-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: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9147225853\)
\(L(\frac12)\) \(\approx\) \(0.9147225853\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.109135943420475501148330172576, −8.466966762762631626376120802927, −8.259742304248469417798217552969, −7.999880118589760327918723378291, −7.54983137185411787538601788228, −7.02853815781715657927744891690, −6.69225628771676496979331775382, −6.45105118184684078240211004390, −5.67031966948073120559224563017, −5.51328388472852407928465851006, −5.02268489259954519055553994057, −4.96283613240339479854741446883, −4.00584929313426149915755222798, −4.00306158976916231834111828097, −3.37781212727533182471597819264, −2.82082217816716706850600038464, −2.39142595406371676567320254155, −1.93527283066148369654845880233, −1.11106422869525614238707343327, −0.33590614346143575041293761890, 0.33590614346143575041293761890, 1.11106422869525614238707343327, 1.93527283066148369654845880233, 2.39142595406371676567320254155, 2.82082217816716706850600038464, 3.37781212727533182471597819264, 4.00306158976916231834111828097, 4.00584929313426149915755222798, 4.96283613240339479854741446883, 5.02268489259954519055553994057, 5.51328388472852407928465851006, 5.67031966948073120559224563017, 6.45105118184684078240211004390, 6.69225628771676496979331775382, 7.02853815781715657927744891690, 7.54983137185411787538601788228, 7.999880118589760327918723378291, 8.259742304248469417798217552969, 8.466966762762631626376120802927, 9.109135943420475501148330172576

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