Properties

Label 4-1815e2-1.1-c1e2-0-14
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  − 2·4-s − 3·5-s + 9-s + 6·20-s − 3·23-s + 4·25-s + 4·31-s − 2·36-s + 6·37-s − 3·45-s − 9·47-s − 4·49-s − 18·53-s + 3·59-s + 8·64-s + 24·67-s − 3·71-s + 81-s − 3·89-s + 6·92-s − 8·100-s + 6·103-s − 3·113-s + 9·115-s − 8·124-s + 3·125-s + 127-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s + 1/3·9-s + 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.718·31-s − 1/3·36-s + 0.986·37-s − 0.447·45-s − 1.31·47-s − 4/7·49-s − 2.47·53-s + 0.390·59-s + 64-s + 2.93·67-s − 0.356·71-s + 1/9·81-s − 0.317·89-s + 0.625·92-s − 4/5·100-s + 0.591·103-s − 0.282·113-s + 0.839·115-s − 0.718·124-s + 0.268·125-s + 0.0887·127-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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 119 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.47940989837764363954032281192, −6.80263715871571754171837420106, −6.61652829171063868440110633063, −6.12057939698499850735830881946, −5.55845115780048886607875656061, −4.90018075076443130021167442427, −4.74526111593017959386003359162, −4.36344754882561200494685590013, −3.78539727127252690371054728720, −3.59817831642777593627469229477, −2.96785743114780570698224297243, −2.31969335712321486534787457423, −1.54678956680124712333616942355, −0.70948635539042139653477101799, 0, 0.70948635539042139653477101799, 1.54678956680124712333616942355, 2.31969335712321486534787457423, 2.96785743114780570698224297243, 3.59817831642777593627469229477, 3.78539727127252690371054728720, 4.36344754882561200494685590013, 4.74526111593017959386003359162, 4.90018075076443130021167442427, 5.55845115780048886607875656061, 6.12057939698499850735830881946, 6.61652829171063868440110633063, 6.80263715871571754171837420106, 7.47940989837764363954032281192

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