Properties

Label 4-363e2-1.1-c1e2-0-2
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
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  − 2·3-s + 3·4-s + 9-s − 6·12-s + 5·16-s + 12·23-s − 6·25-s + 4·27-s − 2·31-s + 3·36-s + 12·37-s − 6·47-s − 10·48-s − 8·49-s + 18·53-s + 3·64-s − 10·67-s − 24·69-s − 6·71-s + 12·75-s − 11·81-s + 6·89-s + 36·92-s + 4·93-s + 4·97-s − 18·100-s − 8·103-s + ⋯
L(s)  = 1  − 1.15·3-s + 3/2·4-s + 1/3·9-s − 1.73·12-s + 5/4·16-s + 2.50·23-s − 6/5·25-s + 0.769·27-s − 0.359·31-s + 1/2·36-s + 1.97·37-s − 0.875·47-s − 1.44·48-s − 8/7·49-s + 2.47·53-s + 3/8·64-s − 1.22·67-s − 2.88·69-s − 0.712·71-s + 1.38·75-s − 1.22·81-s + 0.635·89-s + 3.75·92-s + 0.414·93-s + 0.406·97-s − 9/5·100-s − 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544738568\)
\(L(\frac12)\) \(\approx\) \(1.544738568\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.391123376615524289634214778394, −8.988163437873413465273261306041, −8.301479934497777032075454779455, −7.70965142943847156216320356415, −7.23712065372103626215323618313, −6.86050060520135022250260291593, −6.38936219009129754588503267840, −5.88246969543650610358824802805, −5.52973647353527519475543324561, −4.86073842749873988463326706974, −4.27394988321841243924919372259, −3.24666884694853708905752948824, −2.79859931977348035552662373493, −1.93572664561895893044784704738, −0.944613204620489295494413365596, 0.944613204620489295494413365596, 1.93572664561895893044784704738, 2.79859931977348035552662373493, 3.24666884694853708905752948824, 4.27394988321841243924919372259, 4.86073842749873988463326706974, 5.52973647353527519475543324561, 5.88246969543650610358824802805, 6.38936219009129754588503267840, 6.86050060520135022250260291593, 7.23712065372103626215323618313, 7.70965142943847156216320356415, 8.301479934497777032075454779455, 8.988163437873413465273261306041, 9.391123376615524289634214778394

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