Properties

Label 4-1815e2-1.1-c1e2-0-4
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  − 3-s + 3·4-s − 5-s − 2·9-s − 3·12-s + 15-s + 5·16-s − 3·20-s − 4·25-s + 5·27-s + 31-s − 6·36-s − 14·37-s + 2·45-s + 4·47-s − 5·48-s − 12·49-s − 11·53-s + 12·59-s + 3·60-s + 3·64-s + 25·67-s − 3·71-s + 4·75-s − 5·80-s + 81-s + 11·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 3/2·4-s − 0.447·5-s − 2/3·9-s − 0.866·12-s + 0.258·15-s + 5/4·16-s − 0.670·20-s − 4/5·25-s + 0.962·27-s + 0.179·31-s − 36-s − 2.30·37-s + 0.298·45-s + 0.583·47-s − 0.721·48-s − 1.71·49-s − 1.51·53-s + 1.56·59-s + 0.387·60-s + 3/8·64-s + 3.05·67-s − 0.356·71-s + 0.461·75-s − 0.559·80-s + 1/9·81-s + 1.16·89-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

\(L(1)\) \(\approx\) \(1.615552834\)
\(L(\frac12)\) \(\approx\) \(1.615552834\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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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

−7.29316741837956753091988965291, −7.10174299491743267534618588539, −6.66629708925143024020852132943, −6.30133620202751594821626267587, −5.98483625328929459804930880660, −5.48779296887172550633474766576, −5.03523358428498719165312586766, −4.75259852847506779695144876853, −3.87031752410699741952415229388, −3.50949129808201613409097978605, −3.13619202456357364575031659306, −2.43696212496859902445058364990, −2.05094264714191023505924570180, −1.43678046746513523960869698962, −0.47052767748480823200973899330, 0.47052767748480823200973899330, 1.43678046746513523960869698962, 2.05094264714191023505924570180, 2.43696212496859902445058364990, 3.13619202456357364575031659306, 3.50949129808201613409097978605, 3.87031752410699741952415229388, 4.75259852847506779695144876853, 5.03523358428498719165312586766, 5.48779296887172550633474766576, 5.98483625328929459804930880660, 6.30133620202751594821626267587, 6.66629708925143024020852132943, 7.10174299491743267534618588539, 7.29316741837956753091988965291

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