Properties

Label 4-1805e2-1.1-c1e2-0-5
Degree $4$
Conductor $3258025$
Sign $1$
Analytic cond. $207.734$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·5-s − 4·7-s − 6·9-s − 3·16-s − 4·17-s − 2·20-s + 12·23-s + 3·25-s − 4·28-s + 8·35-s − 6·36-s − 12·43-s + 12·45-s − 4·47-s − 2·49-s − 20·61-s + 24·63-s − 7·64-s − 4·68-s + 28·73-s + 6·80-s + 27·81-s + 28·83-s + 8·85-s + 12·92-s + 3·100-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s − 1.51·7-s − 2·9-s − 3/4·16-s − 0.970·17-s − 0.447·20-s + 2.50·23-s + 3/5·25-s − 0.755·28-s + 1.35·35-s − 36-s − 1.82·43-s + 1.78·45-s − 0.583·47-s − 2/7·49-s − 2.56·61-s + 3.02·63-s − 7/8·64-s − 0.485·68-s + 3.27·73-s + 0.670·80-s + 3·81-s + 3.07·83-s + 0.867·85-s + 1.25·92-s + 3/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3258025\)    =    \(5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(207.734\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1805} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3258025,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad5$C_1$ \( ( 1 + T )^{2} \)
19 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
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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

−9.014339022682654525788748670100, −8.897749114707035393122973926869, −8.268810609789432089132763144540, −7.900675296222237214344109636738, −7.59114460523762221657093265967, −6.84033795625724349746851931118, −6.60416628536008257449352409647, −6.47890268486684764787876703445, −6.12713122160770754987246398695, −5.27056767614506292425312527395, −4.96142906670360965911811463386, −4.79290576208334763260373758918, −3.76543315121891478392405415145, −3.48293467799567444191004825750, −3.10867401144006853271230976319, −2.65466888321512335124647307280, −2.31254564290464559679453706839, −1.20098488214232486959183582723, 0, 0, 1.20098488214232486959183582723, 2.31254564290464559679453706839, 2.65466888321512335124647307280, 3.10867401144006853271230976319, 3.48293467799567444191004825750, 3.76543315121891478392405415145, 4.79290576208334763260373758918, 4.96142906670360965911811463386, 5.27056767614506292425312527395, 6.12713122160770754987246398695, 6.47890268486684764787876703445, 6.60416628536008257449352409647, 6.84033795625724349746851931118, 7.59114460523762221657093265967, 7.900675296222237214344109636738, 8.268810609789432089132763144540, 8.897749114707035393122973926869, 9.014339022682654525788748670100

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