Properties

Label 4-3330e2-1.1-c1e2-0-20
Degree $4$
Conductor $11088900$
Sign $1$
Analytic cond. $707.037$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 4·5-s + 4·8-s − 8·10-s + 6·11-s − 8·13-s + 5·16-s − 14·17-s − 12·20-s + 12·22-s − 12·23-s + 11·25-s − 16·26-s + 6·32-s − 28·34-s + 12·37-s − 16·40-s − 14·41-s + 2·43-s + 18·44-s − 24·46-s + 5·49-s + 22·50-s − 24·52-s − 24·55-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s + 1.80·11-s − 2.21·13-s + 5/4·16-s − 3.39·17-s − 2.68·20-s + 2.55·22-s − 2.50·23-s + 11/5·25-s − 3.13·26-s + 1.06·32-s − 4.80·34-s + 1.97·37-s − 2.52·40-s − 2.18·41-s + 0.304·43-s + 2.71·44-s − 3.53·46-s + 5/7·49-s + 3.11·50-s − 3.32·52-s − 3.23·55-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11088900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(707.037\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3330} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 11088900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_2$ \( 1 - 12 T + p T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−8.359868199944863199021240091301, −7.956555329998575745283673494655, −7.41021361158273499063481624693, −7.35996393949692343753206835153, −6.76706723812939217514339957131, −6.69920253220864371515547146483, −6.16840648165814891284467185500, −5.96997046907787308616731441224, −5.09850215650327575403620761412, −4.74289753763430731219637328693, −4.47838947328881990374318444354, −4.21866124720168452169509188987, −3.85232890071935981437975915187, −3.67240067800049302071045514792, −2.67003249263391527849169126892, −2.58452729523013995160084059365, −2.03598916193688812775897295309, −1.38628598957560626554783562742, 0, 0, 1.38628598957560626554783562742, 2.03598916193688812775897295309, 2.58452729523013995160084059365, 2.67003249263391527849169126892, 3.67240067800049302071045514792, 3.85232890071935981437975915187, 4.21866124720168452169509188987, 4.47838947328881990374318444354, 4.74289753763430731219637328693, 5.09850215650327575403620761412, 5.96997046907787308616731441224, 6.16840648165814891284467185500, 6.69920253220864371515547146483, 6.76706723812939217514339957131, 7.35996393949692343753206835153, 7.41021361158273499063481624693, 7.956555329998575745283673494655, 8.359868199944863199021240091301

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