Properties

Label 4-3822e2-1.1-c1e2-0-6
Degree $4$
Conductor $14607684$
Sign $1$
Analytic cond. $931.398$
Root an. cond. $5.52438$
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  − 2·2-s + 2·3-s + 3·4-s + 5-s − 4·6-s − 4·8-s + 3·9-s − 2·10-s + 3·11-s + 6·12-s + 2·13-s + 2·15-s + 5·16-s + 7·17-s − 6·18-s − 3·19-s + 3·20-s − 6·22-s − 3·23-s − 8·24-s + 5·25-s − 4·26-s + 4·27-s + 9·29-s − 4·30-s − 16·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s − 1.41·8-s + 9-s − 0.632·10-s + 0.904·11-s + 1.73·12-s + 0.554·13-s + 0.516·15-s + 5/4·16-s + 1.69·17-s − 1.41·18-s − 0.688·19-s + 0.670·20-s − 1.27·22-s − 0.625·23-s − 1.63·24-s + 25-s − 0.784·26-s + 0.769·27-s + 1.67·29-s − 0.730·30-s − 2.87·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14607684\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(931.398\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14607684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.498370692\)
\(L(\frac12)\) \(\approx\) \(3.498370692\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
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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

−8.632718934712415615076700767581, −8.540468440032149254502933065635, −7.974300782950850933387185747782, −7.68126026754969796509913755469, −7.47420219748482946486561644420, −6.88517958231511771567876718046, −6.54710810803300730587817640424, −6.47840322014192927241071559986, −5.72665115840861795775300775652, −5.42521124514534574699953283675, −5.10741852580786120667089600485, −4.16009129305030510463352179807, −3.95735505861470362264923657030, −3.53998009425476247684296362885, −3.07523652984454287831156098822, −2.62025861886281616824784174913, −2.01144168489934844065351739579, −1.79978323713822224518898510137, −1.06710519062030289478825316279, −0.75017728115126000794265505029, 0.75017728115126000794265505029, 1.06710519062030289478825316279, 1.79978323713822224518898510137, 2.01144168489934844065351739579, 2.62025861886281616824784174913, 3.07523652984454287831156098822, 3.53998009425476247684296362885, 3.95735505861470362264923657030, 4.16009129305030510463352179807, 5.10741852580786120667089600485, 5.42521124514534574699953283675, 5.72665115840861795775300775652, 6.47840322014192927241071559986, 6.54710810803300730587817640424, 6.88517958231511771567876718046, 7.47420219748482946486561644420, 7.68126026754969796509913755469, 7.974300782950850933387185747782, 8.540468440032149254502933065635, 8.632718934712415615076700767581

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