Properties

Label 4-1815e2-1.1-c1e2-0-34
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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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 + 3·25-s − 4·27-s − 16·31-s − 3·36-s − 4·37-s − 6·45-s + 6·48-s − 14·49-s − 12·53-s + 24·59-s − 4·60-s + 7·64-s − 16·67-s + 24·71-s − 6·75-s + 6·80-s + 5·81-s − 12·89-s + 32·93-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 + 3/5·25-s − 0.769·27-s − 2.87·31-s − 1/2·36-s − 0.657·37-s − 0.894·45-s + 0.866·48-s − 2·49-s − 1.64·53-s + 3.12·59-s − 0.516·60-s + 7/8·64-s − 1.95·67-s + 2.84·71-s − 0.692·75-s + 0.670·80-s + 5/9·81-s − 1.27·89-s + 3.31·93-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: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.967158660851930466237792642626, −8.820011240823149749756589023974, −8.083568253183071223840268781850, −8.046036996222203290564395408235, −7.38858886797075228904562480866, −6.89974843915113207165478882704, −6.88247408248040237126871603554, −6.33436747418416840785140823830, −5.65189285811065313164211046678, −5.46071292649735554081986056497, −4.82510184431625082527723553300, −4.81308268959775627048620141621, −3.92156562137037026080916460033, −3.86426253102289288200164936510, −3.33025280079733358554184540924, −2.50225461319321110788877891944, −1.82097063686650375861076681109, −1.17998737446998746432223357651, 0, 0, 1.17998737446998746432223357651, 1.82097063686650375861076681109, 2.50225461319321110788877891944, 3.33025280079733358554184540924, 3.86426253102289288200164936510, 3.92156562137037026080916460033, 4.81308268959775627048620141621, 4.82510184431625082527723553300, 5.46071292649735554081986056497, 5.65189285811065313164211046678, 6.33436747418416840785140823830, 6.88247408248040237126871603554, 6.89974843915113207165478882704, 7.38858886797075228904562480866, 8.046036996222203290564395408235, 8.083568253183071223840268781850, 8.820011240823149749756589023974, 8.967158660851930466237792642626

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