Properties

Label 4-1815e2-1.1-c1e2-0-40
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 2·5-s − 4·7-s + 3·9-s − 2·12-s − 4·13-s − 4·15-s − 3·16-s − 4·19-s + 2·20-s − 8·21-s + 3·25-s + 4·27-s + 4·28-s − 8·31-s + 8·35-s − 3·36-s + 4·37-s − 8·39-s − 4·43-s − 6·45-s − 6·48-s − 2·49-s + 4·52-s − 12·53-s − 8·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 9-s − 0.577·12-s − 1.10·13-s − 1.03·15-s − 3/4·16-s − 0.917·19-s + 0.447·20-s − 1.74·21-s + 3/5·25-s + 0.769·27-s + 0.755·28-s − 1.43·31-s + 1.35·35-s − 1/2·36-s + 0.657·37-s − 1.28·39-s − 0.609·43-s − 0.894·45-s − 0.866·48-s − 2/7·49-s + 0.554·52-s − 1.64·53-s − 1.05·57-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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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

−9.012646344264965441331115386482, −8.852164758348461743985909706088, −8.163900095235817963358518439646, −8.013413489399351804721201627120, −7.64619824307247359787219502145, −6.99571487166376361457011261989, −6.79993068119665872589232644652, −6.57172900199190558672907389573, −5.90398600771120009926611340783, −5.31363866241500678720553305179, −4.72565192280137514755377836435, −4.51984147268459278099873628979, −3.78473576132666394055654531978, −3.73464995750232537730113765193, −3.10289076230485859240868530454, −2.64695855207896765958810260536, −2.22860000122942472953322067492, −1.40086716744547073554651145561, 0, 0, 1.40086716744547073554651145561, 2.22860000122942472953322067492, 2.64695855207896765958810260536, 3.10289076230485859240868530454, 3.73464995750232537730113765193, 3.78473576132666394055654531978, 4.51984147268459278099873628979, 4.72565192280137514755377836435, 5.31363866241500678720553305179, 5.90398600771120009926611340783, 6.57172900199190558672907389573, 6.79993068119665872589232644652, 6.99571487166376361457011261989, 7.64619824307247359787219502145, 8.013413489399351804721201627120, 8.163900095235817963358518439646, 8.852164758348461743985909706088, 9.012646344264965441331115386482

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