Properties

Label 4-1815e2-1.1-c1e2-0-15
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s − 2·5-s + 3·9-s − 6·12-s − 4·15-s + 5·16-s + 6·20-s + 8·23-s + 3·25-s + 4·27-s + 16·31-s − 9·36-s − 12·37-s − 6·45-s + 16·47-s + 10·48-s − 10·49-s + 12·53-s + 12·60-s − 3·64-s + 24·67-s + 16·69-s + 16·71-s + 6·75-s − 10·80-s + 5·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s − 0.894·5-s + 9-s − 1.73·12-s − 1.03·15-s + 5/4·16-s + 1.34·20-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 2.87·31-s − 3/2·36-s − 1.97·37-s − 0.894·45-s + 2.33·47-s + 1.44·48-s − 1.42·49-s + 1.64·53-s + 1.54·60-s − 3/8·64-s + 2.93·67-s + 1.92·69-s + 1.89·71-s + 0.692·75-s − 1.11·80-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: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.382360294\)
\(L(\frac12)\) \(\approx\) \(2.382360294\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−7.69344890685142555942945577499, −7.12224324890981759228654811108, −6.81358583409942750681700792930, −6.53038417539122512528573963592, −5.65462395656023150861273258602, −5.12106858409169630942967149663, −4.97190009229779476100580095986, −4.31338610647448896952875925848, −4.18139470284158080833278340409, −3.42381310446072218205035911280, −3.40996389133644129274747060882, −2.64207358991310319726851351291, −2.15956392843062891904606784720, −1.01147500994564153503812445251, −0.71020369206114523558950625222, 0.71020369206114523558950625222, 1.01147500994564153503812445251, 2.15956392843062891904606784720, 2.64207358991310319726851351291, 3.40996389133644129274747060882, 3.42381310446072218205035911280, 4.18139470284158080833278340409, 4.31338610647448896952875925848, 4.97190009229779476100580095986, 5.12106858409169630942967149663, 5.65462395656023150861273258602, 6.53038417539122512528573963592, 6.81358583409942750681700792930, 7.12224324890981759228654811108, 7.69344890685142555942945577499

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