Properties

Label 4-46208-1.1-c1e2-0-0
Degree $4$
Conductor $46208$
Sign $1$
Analytic cond. $2.94626$
Root an. cond. $1.31014$
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-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s + 4·11-s − 2·12-s + 16-s + 6·17-s − 3·18-s − 2·19-s + 4·22-s − 2·24-s + 6·25-s + 14·27-s + 32-s − 8·33-s + 6·34-s − 3·36-s − 2·38-s − 16·41-s + 8·43-s + 4·44-s − 2·48-s − 5·49-s + 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s + 1.20·11-s − 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.458·19-s + 0.852·22-s − 0.408·24-s + 6/5·25-s + 2.69·27-s + 0.176·32-s − 1.39·33-s + 1.02·34-s − 1/2·36-s − 0.324·38-s − 2.49·41-s + 1.21·43-s + 0.603·44-s − 0.288·48-s − 5/7·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46208\)    =    \(2^{7} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2.94626\)
Root analytic conductor: \(1.31014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46208,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364795230\)
\(L(\frac12)\) \(\approx\) \(1.364795230\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−10.16790346439735025567145813035, −9.958642910212366556628274508824, −9.089395378892366384411617506988, −8.408486909011940238981299566497, −8.336890921646432967363618675577, −7.25038064772684706053554512263, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −5.70311644291868843897751049539, −5.28069074518443602737935578596, −4.89838836690724431377597203004, −3.86057015395193111701613139713, −3.35219753314436267860210658260, −2.45747388367035991238545338827, −1.03106032747078173831338248002, 1.03106032747078173831338248002, 2.45747388367035991238545338827, 3.35219753314436267860210658260, 3.86057015395193111701613139713, 4.89838836690724431377597203004, 5.28069074518443602737935578596, 5.70311644291868843897751049539, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 7.25038064772684706053554512263, 8.336890921646432967363618675577, 8.408486909011940238981299566497, 9.089395378892366384411617506988, 9.958642910212366556628274508824, 10.16790346439735025567145813035

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