Properties

Label 4-304e2-1.1-c1e2-0-7
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 3·9-s + 8·11-s + 5·13-s − 3·15-s + 5·17-s − 8·19-s − 23-s + 5·25-s + 8·27-s − 3·29-s − 8·31-s + 8·33-s + 4·37-s + 5·39-s + 5·41-s − 11·43-s − 9·45-s − 5·47-s − 14·49-s + 5·51-s + 9·53-s − 24·55-s − 8·57-s + 13·59-s + 61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 9-s + 2.41·11-s + 1.38·13-s − 0.774·15-s + 1.21·17-s − 1.83·19-s − 0.208·23-s + 25-s + 1.53·27-s − 0.557·29-s − 1.43·31-s + 1.39·33-s + 0.657·37-s + 0.800·39-s + 0.780·41-s − 1.67·43-s − 1.34·45-s − 0.729·47-s − 2·49-s + 0.700·51-s + 1.23·53-s − 3.23·55-s − 1.05·57-s + 1.69·59-s + 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867193608\)
\(L(\frac12)\) \(\approx\) \(1.867193608\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.73642150993963399831469577158, −11.58697672930021961809791810809, −11.15891910130723296563063500809, −10.64707559479990751395030112701, −9.986088556969341745235437774794, −9.614369621802696501389115401608, −8.895922924224607761627142245965, −8.692836682834387140070136896912, −8.194093678871529073832657475720, −7.79438621243966777603425850245, −6.98457395880764282566472946613, −6.72668074657122062497307922331, −6.29734339844758105692908509347, −5.50674024609109312674876864737, −4.43120971616653911414314481139, −4.18010988530899099844113085465, −3.58997165969822650202113996921, −3.38126928343609139256602274873, −1.87332807902193277179806298658, −1.15031396200215619900310020609, 1.15031396200215619900310020609, 1.87332807902193277179806298658, 3.38126928343609139256602274873, 3.58997165969822650202113996921, 4.18010988530899099844113085465, 4.43120971616653911414314481139, 5.50674024609109312674876864737, 6.29734339844758105692908509347, 6.72668074657122062497307922331, 6.98457395880764282566472946613, 7.79438621243966777603425850245, 8.194093678871529073832657475720, 8.692836682834387140070136896912, 8.895922924224607761627142245965, 9.614369621802696501389115401608, 9.986088556969341745235437774794, 10.64707559479990751395030112701, 11.15891910130723296563063500809, 11.58697672930021961809791810809, 11.73642150993963399831469577158

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