Properties

Label 4-3800e2-1.1-c1e2-0-1
Degree $4$
Conductor $14440000$
Sign $1$
Analytic cond. $920.706$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 8·11-s − 2·19-s + 12·29-s − 16·31-s − 4·41-s + 14·49-s + 16·59-s + 4·61-s − 16·71-s + 8·79-s − 5·81-s + 36·89-s − 16·99-s + 20·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.41·11-s − 0.458·19-s + 2.22·29-s − 2.87·31-s − 0.624·41-s + 2·49-s + 2.08·59-s + 0.512·61-s − 1.89·71-s + 0.900·79-s − 5/9·81-s + 3.81·89-s − 1.60·99-s + 1.99·101-s + 0.383·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14440000\)    =    \(2^{6} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(920.706\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564318055\)
\(L(\frac12)\) \(\approx\) \(1.564318055\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−8.600490612133485140072493170118, −8.368594485502377344560929485343, −7.79429008931522625111058771492, −7.67473378577160993432617844906, −7.19083435255323682069450919347, −7.03026440298664264605205141397, −6.53129176987406667258668231547, −5.92802227178778727402680733423, −5.74021814745134973998248861979, −5.25087172880662677657622445678, −4.81891976734193592567706230130, −4.79337381446210053928085388874, −3.99008239393249547241255919867, −3.72339528297593534327338333426, −3.15642961271282818174487236214, −2.70418186520687253034719450080, −2.09514546165216426893649880899, −2.06889194092672978034692285573, −1.02663657397353437532918763041, −0.40693001706152089466805959728, 0.40693001706152089466805959728, 1.02663657397353437532918763041, 2.06889194092672978034692285573, 2.09514546165216426893649880899, 2.70418186520687253034719450080, 3.15642961271282818174487236214, 3.72339528297593534327338333426, 3.99008239393249547241255919867, 4.79337381446210053928085388874, 4.81891976734193592567706230130, 5.25087172880662677657622445678, 5.74021814745134973998248861979, 5.92802227178778727402680733423, 6.53129176987406667258668231547, 7.03026440298664264605205141397, 7.19083435255323682069450919347, 7.67473378577160993432617844906, 7.79429008931522625111058771492, 8.368594485502377344560929485343, 8.600490612133485140072493170118

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