Properties

Label 4-68992-1.1-c1e2-0-13
Degree $4$
Conductor $68992$
Sign $-1$
Analytic cond. $4.39898$
Root an. cond. $1.44823$
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 + 2·7-s − 8-s − 2·9-s − 3·11-s − 6·13-s − 2·14-s + 16-s + 2·18-s + 3·22-s − 6·25-s + 6·26-s + 2·28-s + 10·31-s − 32-s − 2·36-s − 8·43-s − 3·44-s − 2·47-s − 3·49-s + 6·50-s − 6·52-s − 2·56-s − 2·61-s − 10·62-s − 4·63-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.639·22-s − 6/5·25-s + 1.17·26-s + 0.377·28-s + 1.79·31-s − 0.176·32-s − 1/3·36-s − 1.21·43-s − 0.452·44-s − 0.291·47-s − 3/7·49-s + 0.848·50-s − 0.832·52-s − 0.267·56-s − 0.256·61-s − 1.27·62-s − 0.503·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(68992\)    =    \(2^{7} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(4.39898\)
Root analytic conductor: \(1.44823\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 68992,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
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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

−9.725857572552745759519824485673, −9.146006788604155984542555249897, −8.411007959436005369627870397284, −8.058597202344887295756925055545, −7.81404254761651482548107024141, −7.15840591400986470971485779192, −6.61676522267539644957358626308, −5.90321731776482776920945151655, −5.30458596855966292615339386619, −4.85070690878752993752590217718, −4.18711551831326938570131122284, −2.99528178686815833677012693960, −2.58489259966323910147332536821, −1.68655888102428947084093360530, 0, 1.68655888102428947084093360530, 2.58489259966323910147332536821, 2.99528178686815833677012693960, 4.18711551831326938570131122284, 4.85070690878752993752590217718, 5.30458596855966292615339386619, 5.90321731776482776920945151655, 6.61676522267539644957358626308, 7.15840591400986470971485779192, 7.81404254761651482548107024141, 8.058597202344887295756925055545, 8.411007959436005369627870397284, 9.146006788604155984542555249897, 9.725857572552745759519824485673

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