Properties

Label 4-332928-1.1-c1e2-0-7
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 + 4-s − 8-s + 9-s + 2·11-s + 16-s + 2·17-s − 18-s − 2·19-s − 2·22-s + 2·25-s − 32-s − 2·34-s + 36-s + 2·38-s + 18·41-s + 2·44-s − 4·49-s − 2·50-s + 10·59-s + 64-s − 10·67-s + 2·68-s − 72-s + 10·73-s − 2·76-s + 81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s − 0.426·22-s + 2/5·25-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.324·38-s + 2.81·41-s + 0.301·44-s − 4/7·49-s − 0.282·50-s + 1.30·59-s + 1/8·64-s − 1.22·67-s + 0.242·68-s − 0.117·72-s + 1.17·73-s − 0.229·76-s + 1/9·81-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

\(L(1)\) \(\approx\) \(1.390815689\)
\(L(\frac12)\) \(\approx\) \(1.390815689\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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

−8.871747591477723490966823796019, −8.284219306744146302033965153673, −7.83326095338794487804025198121, −7.51589596344692985995331997935, −6.88582590126517908164325867497, −6.57036829620668455053553584045, −5.99377669881420947816614031366, −5.56651040385884547633307572089, −4.89175420348459165822884119569, −4.20896574320434341938862312034, −3.85107108870254959267147611793, −3.01000289628689571131451698315, −2.43876237620700989917098389577, −1.60437373389557375530623529932, −0.801294814325978538714061369619, 0.801294814325978538714061369619, 1.60437373389557375530623529932, 2.43876237620700989917098389577, 3.01000289628689571131451698315, 3.85107108870254959267147611793, 4.20896574320434341938862312034, 4.89175420348459165822884119569, 5.56651040385884547633307572089, 5.99377669881420947816614031366, 6.57036829620668455053553584045, 6.88582590126517908164325867497, 7.51589596344692985995331997935, 7.83326095338794487804025198121, 8.284219306744146302033965153673, 8.871747591477723490966823796019

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