Properties

Label 4-2475e2-1.1-c3e2-0-5
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $21324.6$
Root an. cond. $12.0842$
Motivic weight $3$
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  + 2-s − 7·4-s − 2·7-s − 7·8-s − 22·11-s + 76·13-s − 2·14-s − 7·16-s − 26·17-s − 54·19-s − 22·22-s + 224·23-s + 76·26-s + 14·28-s − 222·29-s − 40·31-s − 71·32-s − 26·34-s + 48·37-s − 54·38-s + 494·41-s + 66·43-s + 154·44-s + 224·46-s − 64·47-s − 650·49-s − 532·52-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 0.107·7-s − 0.309·8-s − 0.603·11-s + 1.62·13-s − 0.0381·14-s − 0.109·16-s − 0.370·17-s − 0.652·19-s − 0.213·22-s + 2.03·23-s + 0.573·26-s + 0.0944·28-s − 1.42·29-s − 0.231·31-s − 0.392·32-s − 0.131·34-s + 0.213·37-s − 0.230·38-s + 1.88·41-s + 0.234·43-s + 0.527·44-s + 0.717·46-s − 0.198·47-s − 1.89·49-s − 1.41·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(21324.6\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6125625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 - T + p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 54 T + 11774 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 112 T + p^{3} T^{2} )^{2} \)
29$D_{4}$ \( 1 + 222 T + 43642 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 48 T + 85910 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 494 T + 198818 T^{2} - 494 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 64 T + 189662 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 84 T + 164350 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 196 T + p^{3} T^{2} )^{2} \)
61$D_{4}$ \( 1 + 1104 T + 736358 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 928 T + 626214 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 456 T + 488494 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 592 T + 341742 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 230 T + 954126 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 348 T + 307798 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 972 T + 1645606 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1184 T + 720510 T^{2} - 1184 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.383432276836020511910748794302, −8.117197849919062745735023717630, −7.57282762032140643523432448834, −7.34143587164286606854414559030, −6.83258633665706171167509444723, −6.30770345289945794771290652168, −5.96993433831302253600058824036, −5.78808743021396520039287505448, −5.00668341585412550275631070630, −4.91646128734486003802521790481, −4.31588972811561743755417176405, −4.16388360016635995681965260437, −3.37217961161974787175086997658, −3.28969113968720517117404041297, −2.66905914185849372913419041955, −2.02559560200012712446118142456, −1.39622412384712228152719190871, −0.999438282770302280045953581331, 0, 0, 0.999438282770302280045953581331, 1.39622412384712228152719190871, 2.02559560200012712446118142456, 2.66905914185849372913419041955, 3.28969113968720517117404041297, 3.37217961161974787175086997658, 4.16388360016635995681965260437, 4.31588972811561743755417176405, 4.91646128734486003802521790481, 5.00668341585412550275631070630, 5.78808743021396520039287505448, 5.96993433831302253600058824036, 6.30770345289945794771290652168, 6.83258633665706171167509444723, 7.34143587164286606854414559030, 7.57282762032140643523432448834, 8.117197849919062745735023717630, 8.383432276836020511910748794302

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