Properties

Label 4-2347884-1.1-c1e2-0-5
Degree $4$
Conductor $2347884$
Sign $-1$
Analytic cond. $149.703$
Root an. cond. $3.49790$
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 − 5-s + 3·9-s + 11-s + 2·12-s + 2·15-s + 16-s + 20-s − 7·23-s − 3·25-s − 4·27-s + 31-s − 2·33-s − 3·36-s + 7·37-s − 44-s − 3·45-s + 13·47-s − 2·48-s + 49-s − 5·53-s − 55-s + 3·59-s − 2·60-s − 64-s + 12·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.447·5-s + 9-s + 0.301·11-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 0.223·20-s − 1.45·23-s − 3/5·25-s − 0.769·27-s + 0.179·31-s − 0.348·33-s − 1/2·36-s + 1.15·37-s − 0.150·44-s − 0.447·45-s + 1.89·47-s − 0.288·48-s + 1/7·49-s − 0.686·53-s − 0.134·55-s + 0.390·59-s − 0.258·60-s − 1/8·64-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2347884\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(149.703\)
Root analytic conductor: \(3.49790\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2347884} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2347884,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
61$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−7.52762712697587657263010644471, −7.00402164374025168755318548697, −6.54214085601683384907620106545, −6.10878976323756887844324767104, −5.77503041166670714267004163067, −5.43400243199392372637399321908, −4.87572181271043075163269651748, −4.28269410667252079460638359096, −4.13789819548286247267062318976, −3.71850034705211522774469739571, −2.93406378582761935887723097576, −2.27134608949972785456222357257, −1.54393518050178245150443581761, −0.791960728109073280055411640713, 0, 0.791960728109073280055411640713, 1.54393518050178245150443581761, 2.27134608949972785456222357257, 2.93406378582761935887723097576, 3.71850034705211522774469739571, 4.13789819548286247267062318976, 4.28269410667252079460638359096, 4.87572181271043075163269651748, 5.43400243199392372637399321908, 5.77503041166670714267004163067, 6.10878976323756887844324767104, 6.54214085601683384907620106545, 7.00402164374025168755318548697, 7.52762712697587657263010644471

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