Properties

Label 4-1815e2-1.1-c1e2-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s + 2·5-s + 9-s − 6·12-s + 4·15-s + 5·16-s − 6·20-s + 4·23-s − 25-s − 4·27-s + 2·31-s − 3·36-s + 12·37-s + 2·45-s + 2·47-s + 10·48-s + 8·49-s + 6·53-s − 12·60-s − 3·64-s + 6·67-s + 8·69-s − 6·71-s − 2·75-s + 10·80-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 0.894·5-s + 1/3·9-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.34·20-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.359·31-s − 1/2·36-s + 1.97·37-s + 0.298·45-s + 0.291·47-s + 1.44·48-s + 8/7·49-s + 0.824·53-s − 1.54·60-s − 3/8·64-s + 0.733·67-s + 0.963·69-s − 0.712·71-s − 0.230·75-s + 1.11·80-s − 1.22·81-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: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.763312357\)
\(L(\frac12)\) \(\approx\) \(2.763312357\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
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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

−7.61903597546549763610329784051, −7.31746964320638785746059580449, −6.69367836026985911925712888957, −6.08630065779101110146645115760, −5.87199295441569638505379699624, −5.37324179889398776145644655774, −4.89847213809135812830557412675, −4.51999187028564821606503198690, −3.98973302902399015605437597128, −3.67532222647555978345005795222, −3.07373473851789991663251978316, −2.50499289404960237862393568019, −2.19657804181307391051180413721, −1.30271825511364110114353963513, −0.63046087289744001639918732362, 0.63046087289744001639918732362, 1.30271825511364110114353963513, 2.19657804181307391051180413721, 2.50499289404960237862393568019, 3.07373473851789991663251978316, 3.67532222647555978345005795222, 3.98973302902399015605437597128, 4.51999187028564821606503198690, 4.89847213809135812830557412675, 5.37324179889398776145644655774, 5.87199295441569638505379699624, 6.08630065779101110146645115760, 6.69367836026985911925712888957, 7.31746964320638785746059580449, 7.61903597546549763610329784051

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