Properties

Label 4-332928-1.1-c1e2-0-5
Degree $4$
Conductor $332928$
Sign $1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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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 + 2·11-s − 2·12-s + 16-s − 2·17-s + 3·18-s + 2·19-s + 2·22-s − 2·24-s − 2·25-s − 4·27-s + 32-s − 4·33-s − 2·34-s + 3·36-s + 2·38-s − 2·41-s + 2·44-s − 2·48-s + 4·49-s − 2·50-s + 4·51-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 + 0.603·11-s − 0.577·12-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.458·19-s + 0.426·22-s − 0.408·24-s − 2/5·25-s − 0.769·27-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/2·36-s + 0.324·38-s − 0.312·41-s + 0.301·44-s − 0.288·48-s + 4/7·49-s − 0.282·50-s + 0.560·51-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.911493331\)
\(L(\frac12)\) \(\approx\) \(1.911493331\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.662729521800054165473286356243, −8.344049562579227957324235441634, −7.56635200723163640501695601125, −7.22325649498138535741220556470, −6.75993263897348929126217713412, −6.36134146339626209476626392546, −5.79554132910688610106433906848, −5.53116156537789937248947005211, −4.82963247234091756070086741696, −4.53391599546078451785601007835, −3.82429851373784428188553306032, −3.44107404301964160417462896476, −2.45972208595800392311165280326, −1.76059468726269949563434649510, −0.78015145560533773288999151306, 0.78015145560533773288999151306, 1.76059468726269949563434649510, 2.45972208595800392311165280326, 3.44107404301964160417462896476, 3.82429851373784428188553306032, 4.53391599546078451785601007835, 4.82963247234091756070086741696, 5.53116156537789937248947005211, 5.79554132910688610106433906848, 6.36134146339626209476626392546, 6.75993263897348929126217713412, 7.22325649498138535741220556470, 7.56635200723163640501695601125, 8.344049562579227957324235441634, 8.662729521800054165473286356243

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