Properties

Label 4-1815e2-1.1-c1e2-0-41
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 − 4·4-s − 2·5-s + 3·9-s − 8·12-s − 4·15-s + 12·16-s + 8·20-s − 12·23-s + 3·25-s + 4·27-s − 2·31-s − 12·36-s − 14·37-s − 6·45-s + 24·48-s − 13·49-s − 12·53-s − 24·59-s + 16·60-s − 32·64-s − 14·67-s − 24·69-s + 12·71-s + 6·75-s − 24·80-s + 5·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 0.894·5-s + 9-s − 2.30·12-s − 1.03·15-s + 3·16-s + 1.78·20-s − 2.50·23-s + 3/5·25-s + 0.769·27-s − 0.359·31-s − 2·36-s − 2.30·37-s − 0.894·45-s + 3.46·48-s − 1.85·49-s − 1.64·53-s − 3.12·59-s + 2.06·60-s − 4·64-s − 1.71·67-s − 2.88·69-s + 1.42·71-s + 0.692·75-s − 2.68·80-s + 5/9·81-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: 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−7.59654569582148090602817467428, −6.59523037473790869546358443506, −6.43362309054629922205991853367, −5.61750588322265071162144959368, −5.36747921289838822076634820154, −4.65725991292015173841856541050, −4.51845373560017717550608344934, −3.90838681993568032244631066955, −3.82028048843177336490572449531, −3.11214978990016222560190492173, −2.97589293278646371195021990890, −1.63448908436523374581135103796, −1.58360774879508418147674253120, 0, 0, 1.58360774879508418147674253120, 1.63448908436523374581135103796, 2.97589293278646371195021990890, 3.11214978990016222560190492173, 3.82028048843177336490572449531, 3.90838681993568032244631066955, 4.51845373560017717550608344934, 4.65725991292015173841856541050, 5.36747921289838822076634820154, 5.61750588322265071162144959368, 6.43362309054629922205991853367, 6.59523037473790869546358443506, 7.59654569582148090602817467428

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