Properties

Label 4-675e2-1.1-c2e2-0-9
Degree $4$
Conductor $455625$
Sign $1$
Analytic cond. $338.281$
Root an. cond. $4.28863$
Motivic weight $2$
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  + 3·4-s + 6·7-s − 28·13-s − 7·16-s + 2·19-s + 18·28-s − 42·31-s + 106·37-s − 46·43-s − 71·49-s − 84·52-s − 142·61-s − 69·64-s − 12·67-s + 14·73-s + 6·76-s + 102·79-s − 168·91-s + 286·97-s − 386·103-s + 322·109-s − 42·112-s + 222·121-s − 126·124-s + 127-s + 131-s + 12·133-s + ⋯
L(s)  = 1  + 3/4·4-s + 6/7·7-s − 2.15·13-s − 0.437·16-s + 2/19·19-s + 9/14·28-s − 1.35·31-s + 2.86·37-s − 1.06·43-s − 1.44·49-s − 1.61·52-s − 2.32·61-s − 1.07·64-s − 0.179·67-s + 0.191·73-s + 3/38·76-s + 1.29·79-s − 1.84·91-s + 2.94·97-s − 3.74·103-s + 2.95·109-s − 3/8·112-s + 1.83·121-s − 1.01·124-s + 0.00787·127-s + 0.00763·131-s + 0.0902·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(455625\)    =    \(3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(338.281\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 455625,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.085298283\)
\(L(\frac12)\) \(\approx\) \(2.085298283\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 222 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 498 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 562 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 318 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 21 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 53 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2862 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 1362 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3198 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 258 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 71 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 6702 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 51 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4222 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 1262 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 143 T + p^{2} T^{2} )^{2} \)
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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

−10.84765094349573816357212303845, −9.969055135088156734181315424229, −9.621123621556522837950332225906, −9.411970097632457632200310834353, −8.820808290966484546534507290646, −8.192985890107245584972277370265, −7.76617594573990540446715288679, −7.39835802381566846243887199120, −7.25467687474990038985954903615, −6.35553042658080005063994895899, −6.29202638112779991805864777970, −5.46546180992014601835556036027, −4.92673141531797872894234753372, −4.66857377463115535058490330635, −4.15328317732875621511305572573, −3.16601303820475977722836849564, −2.77647651815720253439690392081, −2.03001986328359106768059245269, −1.73093782297716207134725857736, −0.48882213720974476959149839870, 0.48882213720974476959149839870, 1.73093782297716207134725857736, 2.03001986328359106768059245269, 2.77647651815720253439690392081, 3.16601303820475977722836849564, 4.15328317732875621511305572573, 4.66857377463115535058490330635, 4.92673141531797872894234753372, 5.46546180992014601835556036027, 6.29202638112779991805864777970, 6.35553042658080005063994895899, 7.25467687474990038985954903615, 7.39835802381566846243887199120, 7.76617594573990540446715288679, 8.192985890107245584972277370265, 8.820808290966484546534507290646, 9.411970097632457632200310834353, 9.621123621556522837950332225906, 9.969055135088156734181315424229, 10.84765094349573816357212303845

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