Properties

Label 4-1815e2-1.1-c1e2-0-33
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 + 2·4-s + 3·5-s + 3·9-s − 4·12-s − 6·15-s + 6·20-s − 3·23-s + 4·25-s − 4·27-s − 4·31-s + 6·36-s + 2·37-s + 9·45-s − 3·47-s + 4·49-s − 6·53-s + 3·59-s − 12·60-s − 8·64-s − 4·67-s + 6·69-s − 3·71-s − 8·75-s + 5·81-s + 3·89-s − 6·92-s + ⋯
L(s)  = 1  − 1.15·3-s + 4-s + 1.34·5-s + 9-s − 1.15·12-s − 1.54·15-s + 1.34·20-s − 0.625·23-s + 4/5·25-s − 0.769·27-s − 0.718·31-s + 36-s + 0.328·37-s + 1.34·45-s − 0.437·47-s + 4/7·49-s − 0.824·53-s + 0.390·59-s − 1.54·60-s − 64-s − 0.488·67-s + 0.722·69-s − 0.356·71-s − 0.923·75-s + 5/9·81-s + 0.317·89-s − 0.625·92-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$C_2$ \( 1 - 3 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.13769525321656948639436590464, −6.69243350185303817379573537415, −6.44310983299421548882067057503, −6.11340913414956618458754919580, −5.68819495759312838132936180222, −5.38086424547565149198700647252, −4.98032193823229449699335896154, −4.35263962301717241650736623285, −3.97767135717232869460276552512, −3.20564531800342706767460720185, −2.66888506241133297183477036962, −2.10151938021844476995064373803, −1.71624951086665592459460693802, −1.13230946339803826046642567548, 0, 1.13230946339803826046642567548, 1.71624951086665592459460693802, 2.10151938021844476995064373803, 2.66888506241133297183477036962, 3.20564531800342706767460720185, 3.97767135717232869460276552512, 4.35263962301717241650736623285, 4.98032193823229449699335896154, 5.38086424547565149198700647252, 5.68819495759312838132936180222, 6.11340913414956618458754919580, 6.44310983299421548882067057503, 6.69243350185303817379573537415, 7.13769525321656948639436590464

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