Properties

Label 4-6011e2-1.1-c1e2-0-0
Degree $4$
Conductor $36132121$
Sign $1$
Analytic cond. $2303.81$
Root an. cond. $6.92806$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·4-s − 2·7-s + 6·9-s − 8·12-s − 4·13-s + 4·19-s − 8·21-s − 4·23-s − 2·25-s − 4·27-s + 4·28-s − 8·31-s − 12·36-s − 8·37-s − 16·39-s − 6·41-s + 10·43-s − 8·47-s − 11·49-s + 8·52-s + 16·57-s + 18·59-s + 4·61-s − 12·63-s + 8·64-s + 14·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 4-s − 0.755·7-s + 2·9-s − 2.30·12-s − 1.10·13-s + 0.917·19-s − 1.74·21-s − 0.834·23-s − 2/5·25-s − 0.769·27-s + 0.755·28-s − 1.43·31-s − 2·36-s − 1.31·37-s − 2.56·39-s − 0.937·41-s + 1.52·43-s − 1.16·47-s − 1.57·49-s + 1.10·52-s + 2.11·57-s + 2.34·59-s + 0.512·61-s − 1.51·63-s + 64-s + 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36132121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36132121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36132121\)    =    \(6011^{2}\)
Sign: $1$
Analytic conductor: \(2303.81\)
Root analytic conductor: \(6.92806\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 36132121,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad6011 \( 1+O(T) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 151 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−8.037320087542019907088555948111, −7.80227777642978981045443275430, −7.29086405724718732742521075575, −6.89131336795170439683812530891, −6.83144406496366616624902535991, −6.12411223570190490749209900801, −5.59315384846023000001200001534, −5.33599932400916277552699035183, −5.02718628241489753749585940691, −4.51824382186952005110477732235, −3.92453793980422204078593130716, −3.70934060798752732765978003320, −3.46759832927655197654433517784, −3.13490711696784360345618452452, −2.52817507160489180094504075009, −2.20553243954179432429620854291, −1.97859821165285673217133941978, −1.16736821809428435350262410335, 0, 0, 1.16736821809428435350262410335, 1.97859821165285673217133941978, 2.20553243954179432429620854291, 2.52817507160489180094504075009, 3.13490711696784360345618452452, 3.46759832927655197654433517784, 3.70934060798752732765978003320, 3.92453793980422204078593130716, 4.51824382186952005110477732235, 5.02718628241489753749585940691, 5.33599932400916277552699035183, 5.59315384846023000001200001534, 6.12411223570190490749209900801, 6.83144406496366616624902535991, 6.89131336795170439683812530891, 7.29086405724718732742521075575, 7.80227777642978981045443275430, 8.037320087542019907088555948111

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