Properties

Label 4-3332e2-1.1-c0e2-0-7
Degree $4$
Conductor $11102224$
Sign $1$
Analytic cond. $2.76518$
Root an. cond. $1.28952$
Motivic weight $0$
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  + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11102224\)    =    \(2^{4} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2.76518\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11102224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.904559197\)
\(L(\frac12)\) \(\approx\) \(4.904559197\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$ \( ( 1 - T )^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−8.910798471797272718739241120261, −8.430633300389028200049488011712, −8.048066470631121885324039611231, −7.932166059349246756991103152183, −7.26929135940430910527311972030, −7.04115919702666259433507616079, −6.63573805565276938472088701074, −5.99821582395933511915893744867, −5.95268570667967206761893156753, −5.47196184114791683729290094088, −5.42316652431438731144289533028, −4.84989405626378519022573792706, −4.19642928662406139554867024405, −4.05546430443564320995967653251, −3.50207615625483771515222682221, −3.15546472061134942580040113142, −2.69363529614261304227035350493, −2.10603401765731893028781930880, −2.06773450539787846026591481102, −0.932916404743028582333596457171, 0.932916404743028582333596457171, 2.06773450539787846026591481102, 2.10603401765731893028781930880, 2.69363529614261304227035350493, 3.15546472061134942580040113142, 3.50207615625483771515222682221, 4.05546430443564320995967653251, 4.19642928662406139554867024405, 4.84989405626378519022573792706, 5.42316652431438731144289533028, 5.47196184114791683729290094088, 5.95268570667967206761893156753, 5.99821582395933511915893744867, 6.63573805565276938472088701074, 7.04115919702666259433507616079, 7.26929135940430910527311972030, 7.932166059349246756991103152183, 8.048066470631121885324039611231, 8.430633300389028200049488011712, 8.910798471797272718739241120261

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