Properties

Label 4-7605e2-1.1-c1e2-0-5
Degree $4$
Conductor $57836025$
Sign $1$
Analytic cond. $3687.67$
Root an. cond. $7.79270$
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  + 4·2-s + 8·4-s − 2·5-s + 8·8-s − 8·10-s + 6·11-s − 4·16-s − 16·20-s + 24·22-s + 3·25-s − 32·32-s − 16·40-s + 22·41-s + 8·43-s + 48·44-s − 8·47-s − 49-s + 12·50-s − 12·55-s + 24·59-s + 26·61-s − 64·64-s − 10·71-s + 26·79-s + 8·80-s + 88·82-s + 12·83-s + ⋯
L(s)  = 1  + 2.82·2-s + 4·4-s − 0.894·5-s + 2.82·8-s − 2.52·10-s + 1.80·11-s − 16-s − 3.57·20-s + 5.11·22-s + 3/5·25-s − 5.65·32-s − 2.52·40-s + 3.43·41-s + 1.21·43-s + 7.23·44-s − 1.16·47-s − 1/7·49-s + 1.69·50-s − 1.61·55-s + 3.12·59-s + 3.32·61-s − 8·64-s − 1.18·71-s + 2.92·79-s + 0.894·80-s + 9.71·82-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57836025\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3687.67\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57836025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.31861395\)
\(L(\frac12)\) \(\approx\) \(15.31861395\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 181 T^{2} + p^{2} T^{4} \)
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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

−8.053200353648090983242120639484, −7.40338920510746480309800476729, −7.02107072600789627268707952697, −6.89441128929273641160775269624, −6.40569271404032458512873440630, −6.33398776600490735907830175960, −5.68736306575625854511254355720, −5.61817350694683646713423299377, −5.15524886604877096047680788741, −4.80783884896695767539362879374, −4.32194624646280258307577216615, −4.00047718155192463526339852715, −3.92487742346277226615413609964, −3.75909133339107555969609520747, −3.07512886695855041335256562572, −2.85075288580481486883784449695, −2.19627800604815603375128958152, −2.06812543627327830146096990128, −0.944584940476267486916148904875, −0.67793530494639969641999660350, 0.67793530494639969641999660350, 0.944584940476267486916148904875, 2.06812543627327830146096990128, 2.19627800604815603375128958152, 2.85075288580481486883784449695, 3.07512886695855041335256562572, 3.75909133339107555969609520747, 3.92487742346277226615413609964, 4.00047718155192463526339852715, 4.32194624646280258307577216615, 4.80783884896695767539362879374, 5.15524886604877096047680788741, 5.61817350694683646713423299377, 5.68736306575625854511254355720, 6.33398776600490735907830175960, 6.40569271404032458512873440630, 6.89441128929273641160775269624, 7.02107072600789627268707952697, 7.40338920510746480309800476729, 8.053200353648090983242120639484

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