Properties

Label 4-332928-1.1-c1e2-0-29
Degree $4$
Conductor $332928$
Sign $-1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

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 + 2·12-s + 16-s − 2·17-s − 3·18-s − 8·19-s − 2·24-s − 10·25-s + 4·27-s − 32-s + 2·34-s + 3·36-s + 8·38-s + 12·41-s − 8·43-s + 2·48-s − 10·49-s + 10·50-s − 4·51-s − 4·54-s − 16·57-s − 24·59-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 + 0.577·12-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 1.83·19-s − 0.408·24-s − 2·25-s + 0.769·27-s − 0.176·32-s + 0.342·34-s + 1/2·36-s + 1.29·38-s + 1.87·41-s − 1.21·43-s + 0.288·48-s − 1.42·49-s + 1.41·50-s − 0.560·51-s − 0.544·54-s − 2.11·57-s − 3.12·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(21.2277\)
Root analytic conductor: \(2.14647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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

−8.367508603382134314146293072792, −8.251849824429749124017602038072, −7.64324124944073722179939518360, −7.50641789037179894281138679970, −6.65576551455226941276409559472, −6.30405633057891562651145051326, −5.96956680213085492365415051586, −5.05586136129239314554307726555, −4.37901858420320549126928147503, −4.05372696693836488551925909574, −3.33704793748910042211032942691, −2.70945693922168405804036190722, −2.04126125287275460099535053657, −1.60962373150081650216951780832, 0, 1.60962373150081650216951780832, 2.04126125287275460099535053657, 2.70945693922168405804036190722, 3.33704793748910042211032942691, 4.05372696693836488551925909574, 4.37901858420320549126928147503, 5.05586136129239314554307726555, 5.96956680213085492365415051586, 6.30405633057891562651145051326, 6.65576551455226941276409559472, 7.50641789037179894281138679970, 7.64324124944073722179939518360, 8.251849824429749124017602038072, 8.367508603382134314146293072792

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