Properties

Label 4-21e3-1.1-c1e2-0-0
Degree $4$
Conductor $9261$
Sign $1$
Analytic cond. $0.590489$
Root an. cond. $0.876603$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s + 4·5-s − 7-s + 9-s + 3·12-s − 4·15-s + 5·16-s + 12·17-s − 12·20-s + 21-s + 2·25-s − 27-s + 3·28-s − 4·35-s − 3·36-s + 12·37-s − 4·41-s − 8·43-s + 4·45-s − 5·48-s + 49-s − 12·51-s − 24·59-s + 12·60-s − 63-s − 3·64-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.866·12-s − 1.03·15-s + 5/4·16-s + 2.91·17-s − 2.68·20-s + 0.218·21-s + 2/5·25-s − 0.192·27-s + 0.566·28-s − 0.676·35-s − 1/2·36-s + 1.97·37-s − 0.624·41-s − 1.21·43-s + 0.596·45-s − 0.721·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s + 1.54·60-s − 0.125·63-s − 3/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9261\)    =    \(3^{3} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(0.590489\)
Root analytic conductor: \(0.876603\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9261} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9261,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7969059728\)
\(L(\frac12)\) \(\approx\) \(0.7969059728\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 + T \)
7$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−11.98659319295797721290412051260, −10.77789244508403227045944829665, −10.18331996404873608095625884352, −9.863214598051417092763914791206, −9.457705101580370517739336150901, −9.163045893798789171919999202805, −7.977686193886306753143677629185, −7.81295430367747747098376616892, −6.53484921474790582970192984243, −5.94607665445287604157300008429, −5.48510765590071063887274763802, −5.02709416296405066539370067618, −4.01671863219499060051377207353, −3.05422074105458389777226041971, −1.40786771277750203137263322761, 1.40786771277750203137263322761, 3.05422074105458389777226041971, 4.01671863219499060051377207353, 5.02709416296405066539370067618, 5.48510765590071063887274763802, 5.94607665445287604157300008429, 6.53484921474790582970192984243, 7.81295430367747747098376616892, 7.977686193886306753143677629185, 9.163045893798789171919999202805, 9.457705101580370517739336150901, 9.863214598051417092763914791206, 10.18331996404873608095625884352, 10.77789244508403227045944829665, 11.98659319295797721290412051260

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