Properties

Label 4-1855e2-1.1-c1e2-0-6
Degree $4$
Conductor $3441025$
Sign $-1$
Analytic cond. $219.402$
Root an. cond. $3.84866$
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  − 4·4-s + 2·7-s − 5·9-s − 6·11-s + 10·13-s + 12·16-s + 6·17-s + 25-s − 8·28-s + 6·29-s + 20·36-s + 4·37-s − 20·43-s + 24·44-s + 18·47-s + 3·49-s − 40·52-s + 12·53-s − 10·63-s − 32·64-s − 24·68-s − 12·77-s + 16·81-s − 24·89-s + 20·91-s − 2·97-s + 30·99-s + ⋯
L(s)  = 1  − 2·4-s + 0.755·7-s − 5/3·9-s − 1.80·11-s + 2.77·13-s + 3·16-s + 1.45·17-s + 1/5·25-s − 1.51·28-s + 1.11·29-s + 10/3·36-s + 0.657·37-s − 3.04·43-s + 3.61·44-s + 2.62·47-s + 3/7·49-s − 5.54·52-s + 1.64·53-s − 1.25·63-s − 4·64-s − 2.91·68-s − 1.36·77-s + 16/9·81-s − 2.54·89-s + 2.09·91-s − 0.203·97-s + 3.01·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3441025\)    =    \(5^{2} \cdot 7^{2} \cdot 53^{2}\)
Sign: $-1$
Analytic conductor: \(219.402\)
Root analytic conductor: \(3.84866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3441025} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3441025,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
53$C_2$ \( 1 - 12 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−7.61351728406422952471450498282, −6.88163134068905072786790130896, −6.15165288423916240340816365337, −5.76794291139868174124926619278, −5.47907564040983295563221596641, −5.42303052201623691503191718514, −4.75114534169058008060760688384, −4.40797342389194916116282882235, −3.61689003045514298236484607491, −3.55356439922122982607524011465, −2.98751871897824111763777498176, −2.42156521546257849762441146583, −1.22672678124078466587000691964, −0.956984510621499067485306120141, 0, 0.956984510621499067485306120141, 1.22672678124078466587000691964, 2.42156521546257849762441146583, 2.98751871897824111763777498176, 3.55356439922122982607524011465, 3.61689003045514298236484607491, 4.40797342389194916116282882235, 4.75114534169058008060760688384, 5.42303052201623691503191718514, 5.47907564040983295563221596641, 5.76794291139868174124926619278, 6.15165288423916240340816365337, 6.88163134068905072786790130896, 7.61351728406422952471450498282

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