Properties

Label 4-1815e2-1.1-c1e2-0-24
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  − 3·4-s + 2·5-s + 9-s + 5·16-s − 6·20-s − 12·23-s + 3·25-s + 4·31-s − 3·36-s + 2·45-s + 4·47-s + 6·49-s − 16·53-s − 12·59-s − 3·64-s + 8·67-s + 10·80-s + 81-s − 8·89-s + 36·92-s + 4·97-s − 9·100-s + 16·103-s + 8·113-s − 24·115-s − 12·124-s + 4·125-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.894·5-s + 1/3·9-s + 5/4·16-s − 1.34·20-s − 2.50·23-s + 3/5·25-s + 0.718·31-s − 1/2·36-s + 0.298·45-s + 0.583·47-s + 6/7·49-s − 2.19·53-s − 1.56·59-s − 3/8·64-s + 0.977·67-s + 1.11·80-s + 1/9·81-s − 0.847·89-s + 3.75·92-s + 0.406·97-s − 0.899·100-s + 1.57·103-s + 0.752·113-s − 2.23·115-s − 1.07·124-s + 0.357·125-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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−7.40356201624104244856569236599, −6.84046033789619433597073530611, −6.25454179438831312850661162157, −6.06845307541829334486896406015, −5.65579815452544206052203903413, −5.16896822279346680835295132523, −4.63844401305188353535352714987, −4.43459362689145914513829729877, −3.94087146376237269309386099528, −3.45900947389424748988902415961, −2.86351515773478657205996735173, −2.11704732805279777239310574568, −1.71439451017754662266474088475, −0.873428178317262145023338565327, 0, 0.873428178317262145023338565327, 1.71439451017754662266474088475, 2.11704732805279777239310574568, 2.86351515773478657205996735173, 3.45900947389424748988902415961, 3.94087146376237269309386099528, 4.43459362689145914513829729877, 4.63844401305188353535352714987, 5.16896822279346680835295132523, 5.65579815452544206052203903413, 6.06845307541829334486896406015, 6.25454179438831312850661162157, 6.84046033789619433597073530611, 7.40356201624104244856569236599

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