Properties

Label 4-1815e2-1.1-c1e2-0-37
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 3·5-s + 9-s − 3·16-s + 3·20-s + 4·25-s − 8·31-s + 36-s + 3·45-s + 2·49-s − 12·59-s − 7·64-s − 12·71-s − 9·80-s + 81-s − 18·89-s + 4·100-s − 8·124-s − 3·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s − 24·155-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.34·5-s + 1/3·9-s − 3/4·16-s + 0.670·20-s + 4/5·25-s − 1.43·31-s + 1/6·36-s + 0.447·45-s + 2/7·49-s − 1.56·59-s − 7/8·64-s − 1.42·71-s − 1.00·80-s + 1/9·81-s − 1.90·89-s + 2/5·100-s − 0.718·124-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.0819·149-s + 0.0813·151-s − 1.92·155-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: \(1\)
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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−7.23050463680726890079276924138, −6.85675014000923957999009534498, −6.51100592762843600695651946753, −5.98770498860470517641236611057, −5.74307749942407455731996281143, −5.34468569053018482401502578444, −4.75409093070917884517149339485, −4.40401641897906227937509939922, −3.83312406758686102180441062835, −3.20103971801776995653219856016, −2.71935125721887138945505566949, −2.18682704364323426590934516962, −1.73382695584356415905826384825, −1.26409275326331310959943166221, 0, 1.26409275326331310959943166221, 1.73382695584356415905826384825, 2.18682704364323426590934516962, 2.71935125721887138945505566949, 3.20103971801776995653219856016, 3.83312406758686102180441062835, 4.40401641897906227937509939922, 4.75409093070917884517149339485, 5.34468569053018482401502578444, 5.74307749942407455731996281143, 5.98770498860470517641236611057, 6.51100592762843600695651946753, 6.85675014000923957999009534498, 7.23050463680726890079276924138

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