Properties

Label 4-332928-1.1-c1e2-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 8-s − 3·9-s + 2·10-s + 4·11-s + 16-s + 2·17-s + 3·18-s − 2·20-s − 4·22-s − 6·25-s − 10·29-s − 32-s − 2·34-s − 3·36-s + 10·37-s + 2·40-s + 4·44-s + 6·45-s + 4·47-s + 6·49-s + 6·50-s − 8·55-s + 10·58-s + 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 9-s + 0.632·10-s + 1.20·11-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.447·20-s − 0.852·22-s − 6/5·25-s − 1.85·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.316·40-s + 0.603·44-s + 0.894·45-s + 0.583·47-s + 6/7·49-s + 0.848·50-s − 1.07·55-s + 1.31·58-s + 0.256·61-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: \(1\)
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$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
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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

−8.622136246784754280752018261329, −7.915694487395876219664395608724, −7.72105854024387837794274742842, −7.42286105027581003233803193020, −6.64379118241971121802995961727, −6.29853665636064342378933465254, −5.62188327383372408660495025920, −5.46931551083941803320668058058, −4.41034389635034175302418062871, −3.88483424138462461305067440687, −3.61052086882956948353767975053, −2.77369058996221895298994892895, −2.08863969089520070328586941236, −1.13363909491496290197842393127, 0, 1.13363909491496290197842393127, 2.08863969089520070328586941236, 2.77369058996221895298994892895, 3.61052086882956948353767975053, 3.88483424138462461305067440687, 4.41034389635034175302418062871, 5.46931551083941803320668058058, 5.62188327383372408660495025920, 6.29853665636064342378933465254, 6.64379118241971121802995961727, 7.42286105027581003233803193020, 7.72105854024387837794274742842, 7.915694487395876219664395608724, 8.622136246784754280752018261329

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