Properties

Label 4-56e3-1.1-c1e2-0-31
Degree $4$
Conductor $175616$
Sign $1$
Analytic cond. $11.1974$
Root an. cond. $1.82927$
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  − 4·5-s + 7-s − 6·9-s − 8·11-s − 4·13-s + 2·25-s − 16·31-s − 4·35-s − 8·43-s + 24·45-s + 16·47-s + 49-s + 32·55-s + 12·61-s − 6·63-s + 16·65-s − 8·67-s − 8·77-s + 27·81-s − 4·91-s + 48·99-s − 4·101-s + 32·103-s − 24·107-s + 4·113-s + 24·117-s + 26·121-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.377·7-s − 2·9-s − 2.41·11-s − 1.10·13-s + 2/5·25-s − 2.87·31-s − 0.676·35-s − 1.21·43-s + 3.57·45-s + 2.33·47-s + 1/7·49-s + 4.31·55-s + 1.53·61-s − 0.755·63-s + 1.98·65-s − 0.977·67-s − 0.911·77-s + 3·81-s − 0.419·91-s + 4.82·99-s − 0.398·101-s + 3.15·103-s − 2.32·107-s + 0.376·113-s + 2.21·117-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(175616\)    =    \(2^{9} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(11.1974\)
Root analytic conductor: \(1.82927\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{175616} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 175616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−8.749304120934004278681759852831, −8.078246359041317001571436271675, −7.79292083571821938290976650763, −7.30264187759595822059178656689, −7.26382021666693896035142333427, −5.92791093661634117809390012383, −5.69394971603405508863657314834, −4.98409920426632935260415692638, −4.92225264716551837532790929786, −3.68340574560716411313233602487, −3.60173413237973468362473708205, −2.50561452898491372410349134801, −2.40145156924384381758797773878, 0, 0, 2.40145156924384381758797773878, 2.50561452898491372410349134801, 3.60173413237973468362473708205, 3.68340574560716411313233602487, 4.92225264716551837532790929786, 4.98409920426632935260415692638, 5.69394971603405508863657314834, 5.92791093661634117809390012383, 7.26382021666693896035142333427, 7.30264187759595822059178656689, 7.79292083571821938290976650763, 8.078246359041317001571436271675, 8.749304120934004278681759852831

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