Properties

Label 4-1815e2-1.1-c1e2-0-7
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 2·5-s − 9-s + 5·16-s + 16·19-s − 6·20-s − 25-s + 8·29-s − 16·31-s − 3·36-s − 24·41-s + 2·45-s − 2·49-s + 16·59-s + 3·64-s − 24·71-s + 48·76-s + 16·79-s − 10·80-s + 81-s − 12·89-s − 32·95-s − 3·100-s + 8·101-s − 16·109-s + 24·116-s − 48·124-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s − 1/3·9-s + 5/4·16-s + 3.67·19-s − 1.34·20-s − 1/5·25-s + 1.48·29-s − 2.87·31-s − 1/2·36-s − 3.74·41-s + 0.298·45-s − 2/7·49-s + 2.08·59-s + 3/8·64-s − 2.84·71-s + 5.50·76-s + 1.80·79-s − 1.11·80-s + 1/9·81-s − 1.27·89-s − 3.28·95-s − 0.299·100-s + 0.796·101-s − 1.53·109-s + 2.22·116-s − 4.31·124-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: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.711446192\)
\(L(\frac12)\) \(\approx\) \(2.711446192\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.411667038227991760366920115256, −9.161891937239459487648409481797, −8.514381935759711504909272011063, −8.200405801661613531070445247007, −7.80270855497288390574074728333, −7.28263115555089593740103438497, −7.25108127291580404937172869380, −6.83285221950036323732294950402, −6.47491528293466444118411478083, −5.68801215633961191125092706376, −5.40582086246320653379947184613, −5.27097335197109517589856739534, −4.58133251846812205813242500453, −3.79670714277806958034241819906, −3.40441335206875611386580634459, −3.14672495861360729195195729473, −2.76117907410350819131260758243, −1.76465399518399572789701647427, −1.58297666033820316851832774872, −0.59942514890474629788761004976, 0.59942514890474629788761004976, 1.58297666033820316851832774872, 1.76465399518399572789701647427, 2.76117907410350819131260758243, 3.14672495861360729195195729473, 3.40441335206875611386580634459, 3.79670714277806958034241819906, 4.58133251846812205813242500453, 5.27097335197109517589856739534, 5.40582086246320653379947184613, 5.68801215633961191125092706376, 6.47491528293466444118411478083, 6.83285221950036323732294950402, 7.25108127291580404937172869380, 7.28263115555089593740103438497, 7.80270855497288390574074728333, 8.200405801661613531070445247007, 8.514381935759711504909272011063, 9.161891937239459487648409481797, 9.411667038227991760366920115256

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