Properties

Label 4-1815e2-1.1-c1e2-0-30
Degree $4$
Conductor $3294225$
Sign $-1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·4-s + 5-s + 9-s − 6·12-s − 2·15-s + 5·16-s + 3·20-s − 4·23-s − 4·25-s + 4·27-s − 4·31-s + 3·36-s + 45-s + 4·47-s − 10·48-s − 10·49-s + 18·53-s − 6·60-s + 3·64-s + 8·69-s + 8·75-s + 5·80-s − 11·81-s − 12·92-s + 8·93-s − 12·100-s + ⋯
L(s)  = 1  − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1/3·9-s − 1.73·12-s − 0.516·15-s + 5/4·16-s + 0.670·20-s − 0.834·23-s − 4/5·25-s + 0.769·27-s − 0.718·31-s + 1/2·36-s + 0.149·45-s + 0.583·47-s − 1.44·48-s − 1.42·49-s + 2.47·53-s − 0.774·60-s + 3/8·64-s + 0.963·69-s + 0.923·75-s + 0.559·80-s − 1.22·81-s − 1.25·92-s + 0.829·93-s − 6/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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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

−7.24292596867903156031126681323, −6.85455537810511086070920096884, −6.31467956879432701214176906451, −6.12205185283256534755716218240, −5.75059269610149601501150262068, −5.42980993702063023403755773391, −4.93365822834711710094882709052, −4.37924684181327461533759245710, −3.76270187727379313723657177065, −3.35811893165568626215775057226, −2.56705481076741539590655747712, −2.28013523400158002708655585343, −1.69984557434057821377595788321, −1.04251462893649328175902331756, 0, 1.04251462893649328175902331756, 1.69984557434057821377595788321, 2.28013523400158002708655585343, 2.56705481076741539590655747712, 3.35811893165568626215775057226, 3.76270187727379313723657177065, 4.37924684181327461533759245710, 4.93365822834711710094882709052, 5.42980993702063023403755773391, 5.75059269610149601501150262068, 6.12205185283256534755716218240, 6.31467956879432701214176906451, 6.85455537810511086070920096884, 7.24292596867903156031126681323

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