Properties

Label 4-585e2-1.1-c1e2-0-23
Degree $4$
Conductor $342225$
Sign $-1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
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  − 3·4-s + 4·7-s − 2·13-s + 5·16-s + 12·19-s + 25-s − 12·28-s − 20·31-s − 4·37-s − 16·43-s − 2·49-s + 6·52-s + 4·61-s − 3·64-s − 20·67-s − 12·73-s − 36·76-s + 24·79-s − 8·91-s − 28·97-s − 3·100-s − 8·103-s − 4·109-s + 20·112-s − 6·121-s + 60·124-s + 127-s + ⋯
L(s)  = 1  − 3/2·4-s + 1.51·7-s − 0.554·13-s + 5/4·16-s + 2.75·19-s + 1/5·25-s − 2.26·28-s − 3.59·31-s − 0.657·37-s − 2.43·43-s − 2/7·49-s + 0.832·52-s + 0.512·61-s − 3/8·64-s − 2.44·67-s − 1.40·73-s − 4.12·76-s + 2.70·79-s − 0.838·91-s − 2.84·97-s − 0.299·100-s − 0.788·103-s − 0.383·109-s + 1.88·112-s − 0.545·121-s + 5.38·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.455924495253961688482406234394, −8.069603374513132812350986247673, −7.77509555057385964870974093324, −7.09926588277152631380243196233, −7.01279831140022405844648391421, −5.80203985872581797785224323059, −5.33227627105416183888148278982, −5.17362861953367702283649965368, −4.80291034122253544985154871503, −4.13601102912022186614527931542, −3.48108877573719190338767515003, −3.11999237486850599731326861173, −1.80936597723698845777463745934, −1.37276923078037774007010116691, 0, 1.37276923078037774007010116691, 1.80936597723698845777463745934, 3.11999237486850599731326861173, 3.48108877573719190338767515003, 4.13601102912022186614527931542, 4.80291034122253544985154871503, 5.17362861953367702283649965368, 5.33227627105416183888148278982, 5.80203985872581797785224323059, 7.01279831140022405844648391421, 7.09926588277152631380243196233, 7.77509555057385964870974093324, 8.069603374513132812350986247673, 8.455924495253961688482406234394

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