Properties

Label 4-1815e2-1.1-c1e2-0-29
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 + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s − 3·16-s + 2·20-s + 10·23-s − 25-s − 4·27-s − 2·31-s + 3·36-s − 6·37-s + 6·45-s − 2·47-s + 6·48-s + 8·49-s + 10·53-s − 18·59-s − 4·60-s − 7·64-s − 4·67-s − 20·69-s + 2·75-s − 6·80-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 0.577·12-s − 1.03·15-s − 3/4·16-s + 0.447·20-s + 2.08·23-s − 1/5·25-s − 0.769·27-s − 0.359·31-s + 1/2·36-s − 0.986·37-s + 0.894·45-s − 0.291·47-s + 0.866·48-s + 8/7·49-s + 1.37·53-s − 2.34·59-s − 0.516·60-s − 7/8·64-s − 0.488·67-s − 2.40·69-s + 0.230·75-s − 0.670·80-s + 5/9·81-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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.23473619210511184343399233246, −6.76232375623165558787477433357, −6.47329759783796833964144798890, −6.14351387784493089419951352939, −5.50171559504015289102333086195, −5.38207188464990178748829781459, −4.94008125660946637735343919034, −4.41086721376393163602033352362, −3.97436273141080246034087263155, −3.23878614191063938806341448105, −2.75749348557958634561676664328, −2.16920294421874178207978396532, −1.59081461334933050296525140582, −1.04172320822173386745969461544, 0, 1.04172320822173386745969461544, 1.59081461334933050296525140582, 2.16920294421874178207978396532, 2.75749348557958634561676664328, 3.23878614191063938806341448105, 3.97436273141080246034087263155, 4.41086721376393163602033352362, 4.94008125660946637735343919034, 5.38207188464990178748829781459, 5.50171559504015289102333086195, 6.14351387784493089419951352939, 6.47329759783796833964144798890, 6.76232375623165558787477433357, 7.23473619210511184343399233246

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