Properties

Label 4-105e2-1.1-c1e2-0-3
Degree $4$
Conductor $11025$
Sign $1$
Analytic cond. $0.702963$
Root an. cond. $0.915657$
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  + 4-s − 2·5-s + 4·7-s − 3·9-s − 3·16-s + 12·17-s − 2·20-s + 3·25-s + 4·28-s − 8·35-s − 3·36-s − 4·37-s + 12·41-s − 16·43-s + 6·45-s − 24·47-s + 9·49-s − 24·59-s − 12·63-s − 7·64-s + 16·67-s + 12·68-s + 16·79-s + 6·80-s + 9·81-s − 24·85-s + 12·89-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 1.51·7-s − 9-s − 3/4·16-s + 2.91·17-s − 0.447·20-s + 3/5·25-s + 0.755·28-s − 1.35·35-s − 1/2·36-s − 0.657·37-s + 1.87·41-s − 2.43·43-s + 0.894·45-s − 3.50·47-s + 9/7·49-s − 3.12·59-s − 1.51·63-s − 7/8·64-s + 1.95·67-s + 1.45·68-s + 1.80·79-s + 0.670·80-s + 81-s − 2.60·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.702963\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{105} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082695022\)
\(L(\frac12)\) \(\approx\) \(1.082695022\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−14.44085454038547082654550860093, −13.75691866366674629273059850121, −12.82009214947011327893698412020, −12.26108891185029541558057106982, −11.73201949777095029447336782148, −11.58250802746344809847045416066, −11.02441236922778163076569590742, −10.55467498088450060964845520155, −9.769668520548840763317123178411, −9.150575487050004757995450630884, −8.253625310433564150149755056434, −7.899835355812656162524348600013, −7.78988597368559588077758915153, −6.77936700043949431351337836028, −6.07563044325703369545735779630, −5.05964223157965452541194320082, −4.96741107106739787117608112172, −3.63401856002163203666662374263, −3.02106091710520419118583338891, −1.60623861147945947594658943182, 1.60623861147945947594658943182, 3.02106091710520419118583338891, 3.63401856002163203666662374263, 4.96741107106739787117608112172, 5.05964223157965452541194320082, 6.07563044325703369545735779630, 6.77936700043949431351337836028, 7.78988597368559588077758915153, 7.899835355812656162524348600013, 8.253625310433564150149755056434, 9.150575487050004757995450630884, 9.769668520548840763317123178411, 10.55467498088450060964845520155, 11.02441236922778163076569590742, 11.58250802746344809847045416066, 11.73201949777095029447336782148, 12.26108891185029541558057106982, 12.82009214947011327893698412020, 13.75691866366674629273059850121, 14.44085454038547082654550860093

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