Properties

Degree $4$
Conductor $10682$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 7-s + 8-s + 4·9-s + 14-s + 3·16-s − 4·18-s + 3·23-s + 8·25-s + 28-s + 3·29-s − 3·32-s − 4·36-s + 4·37-s − 11·43-s − 3·46-s − 6·49-s − 8·50-s + 15·53-s − 56-s − 3·58-s − 4·63-s − 5·64-s + 67-s − 9·71-s + 4·72-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 0.267·14-s + 3/4·16-s − 0.942·18-s + 0.625·23-s + 8/5·25-s + 0.188·28-s + 0.557·29-s − 0.530·32-s − 2/3·36-s + 0.657·37-s − 1.67·43-s − 0.442·46-s − 6/7·49-s − 1.13·50-s + 2.06·53-s − 0.133·56-s − 0.393·58-s − 0.503·63-s − 5/8·64-s + 0.122·67-s − 1.06·71-s + 0.471·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10682 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10682 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10682\)    =    \(2 \cdot 7^{2} \cdot 109\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{10682} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10682,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6395679587\)
\(L(\frac12)\) \(\approx\) \(0.6395679587\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
109$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
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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

−11.45015467590693421164346509702, −10.59531048952671103438107516496, −10.14148718376678192266383006025, −9.937215580517584118283960355209, −9.099653024909612921880514011235, −8.815697466748258356959404572848, −8.156693826142832276194917034905, −7.42648111208650801711524598128, −6.89148387018769867154776861331, −6.29813419050289528037616837827, −5.25365473110579765631698501017, −4.66837443786080150871832452155, −3.85055513300718885076242085541, −2.89009034690832333624089485932, −1.24030723994401829440977114502, 1.24030723994401829440977114502, 2.89009034690832333624089485932, 3.85055513300718885076242085541, 4.66837443786080150871832452155, 5.25365473110579765631698501017, 6.29813419050289528037616837827, 6.89148387018769867154776861331, 7.42648111208650801711524598128, 8.156693826142832276194917034905, 8.815697466748258356959404572848, 9.099653024909612921880514011235, 9.937215580517584118283960355209, 10.14148718376678192266383006025, 10.59531048952671103438107516496, 11.45015467590693421164346509702

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