Properties

Label 4-608e2-1.1-c1e2-0-13
Degree $4$
Conductor $369664$
Sign $-1$
Analytic cond. $23.5700$
Root an. cond. $2.20338$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·9-s − 4·11-s + 6·17-s + 2·19-s + 6·25-s − 14·27-s − 8·33-s − 16·41-s − 8·43-s − 5·49-s + 12·51-s + 4·57-s − 30·59-s − 6·67-s + 18·73-s + 12·75-s − 4·81-s + 12·83-s − 4·97-s + 12·99-s + 14·107-s + 28·113-s − 10·121-s − 32·123-s + 127-s − 16·129-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s − 1.20·11-s + 1.45·17-s + 0.458·19-s + 6/5·25-s − 2.69·27-s − 1.39·33-s − 2.49·41-s − 1.21·43-s − 5/7·49-s + 1.68·51-s + 0.529·57-s − 3.90·59-s − 0.733·67-s + 2.10·73-s + 1.38·75-s − 4/9·81-s + 1.31·83-s − 0.406·97-s + 1.20·99-s + 1.35·107-s + 2.63·113-s − 0.909·121-s − 2.88·123-s + 0.0887·127-s − 1.40·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(369664\)    =    \(2^{10} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(23.5700\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369664} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 369664,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.264845552131759886230905508728, −8.199448456904276296960401284970, −7.65960330166497750263390530877, −7.41068992206199799917476516665, −6.52280948162697403172717458153, −6.17646264461499634926610391599, −5.50702009116960948045704530045, −5.01088818916226849614299890353, −4.86338510531971803230901631456, −3.57195691104337905967362169037, −3.27846026315236304587310905813, −3.01072892067414194047812488684, −2.31224961625843070084602752549, −1.49207601742473373563863301755, 0, 1.49207601742473373563863301755, 2.31224961625843070084602752549, 3.01072892067414194047812488684, 3.27846026315236304587310905813, 3.57195691104337905967362169037, 4.86338510531971803230901631456, 5.01088818916226849614299890353, 5.50702009116960948045704530045, 6.17646264461499634926610391599, 6.52280948162697403172717458153, 7.41068992206199799917476516665, 7.65960330166497750263390530877, 8.199448456904276296960401284970, 8.264845552131759886230905508728

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