Properties

Label 4-3115008-1.1-c1e2-0-0
Degree $4$
Conductor $3115008$
Sign $1$
Analytic cond. $198.615$
Root an. cond. $3.75407$
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  − 2·3-s + 3·9-s + 8·11-s − 12·17-s + 12·19-s − 10·25-s − 4·27-s − 16·33-s + 16·41-s + 24·43-s − 10·49-s + 24·51-s − 24·57-s + 4·67-s + 28·73-s + 20·75-s + 5·81-s + 16·83-s + 8·89-s + 28·97-s + 24·99-s − 8·107-s − 20·113-s + 26·121-s − 32·123-s + 127-s − 48·129-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 2.41·11-s − 2.91·17-s + 2.75·19-s − 2·25-s − 0.769·27-s − 2.78·33-s + 2.49·41-s + 3.65·43-s − 1.42·49-s + 3.36·51-s − 3.17·57-s + 0.488·67-s + 3.27·73-s + 2.30·75-s + 5/9·81-s + 1.75·83-s + 0.847·89-s + 2.84·97-s + 2.41·99-s − 0.773·107-s − 1.88·113-s + 2.36·121-s − 2.88·123-s + 0.0887·127-s − 4.22·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3115008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3115008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3115008\)    =    \(2^{11} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(198.615\)
Root analytic conductor: \(3.75407\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3115008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3115008,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.012232477\)
\(L(\frac12)\) \(\approx\) \(2.012232477\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−7.26948453317897544910090398161, −7.16333848096912753102386607562, −6.57091938769509480942354725697, −6.26794152529161447517798404710, −5.98247083912518480972422452649, −5.59232004512023197121827602220, −4.97254580700680097900229302596, −4.48413639840570454203249914552, −4.13958008081477749813044275306, −3.84092207155881713644722109378, −3.29289095796707662296197413885, −2.22728463189050631057902889299, −2.06399639050261723146787386632, −1.02393888089349693967293608434, −0.73510269161810698907992625597, 0.73510269161810698907992625597, 1.02393888089349693967293608434, 2.06399639050261723146787386632, 2.22728463189050631057902889299, 3.29289095796707662296197413885, 3.84092207155881713644722109378, 4.13958008081477749813044275306, 4.48413639840570454203249914552, 4.97254580700680097900229302596, 5.59232004512023197121827602220, 5.98247083912518480972422452649, 6.26794152529161447517798404710, 6.57091938769509480942354725697, 7.16333848096912753102386607562, 7.26948453317897544910090398161

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