Properties

Label 4-546e2-1.1-c1e2-0-52
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 2·3-s − 4-s + 3·9-s − 2·12-s + 6·13-s + 16-s + 14·17-s + 2·23-s + 9·25-s + 4·27-s − 2·29-s − 3·36-s + 12·39-s − 10·43-s + 2·48-s − 49-s + 28·51-s − 6·52-s − 12·53-s − 26·61-s − 64-s − 14·68-s + 4·69-s + 18·75-s − 24·79-s + 5·81-s − 4·87-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s + 3.39·17-s + 0.417·23-s + 9/5·25-s + 0.769·27-s − 0.371·29-s − 1/2·36-s + 1.92·39-s − 1.52·43-s + 0.288·48-s − 1/7·49-s + 3.92·51-s − 0.832·52-s − 1.64·53-s − 3.32·61-s − 1/8·64-s − 1.69·68-s + 0.481·69-s + 2.07·75-s − 2.70·79-s + 5/9·81-s − 0.428·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.165817439\)
\(L(\frac12)\) \(\approx\) \(3.165817439\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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

−10.76478819585551596315768989104, −10.57712839766422400962204337484, −9.909433120286834848958008835898, −9.833323138282610486409897849216, −9.037207096057459242685360680499, −8.912382805672646779752354853771, −8.393621315860024309714270669955, −7.949835094309691188515700603833, −7.61178950661489259753692982695, −7.21011486025012968461358392795, −6.35093675030024605547797105182, −6.08086898664347684779911898670, −5.34465390238234539769941603269, −4.94946932056961628938788272653, −4.27516766227616299875620750773, −3.50561592638930415211628589898, −3.23123754426432929052188179294, −2.95746342164692720974343031467, −1.38390742784676669259892210015, −1.31855408242780957291214467754, 1.31855408242780957291214467754, 1.38390742784676669259892210015, 2.95746342164692720974343031467, 3.23123754426432929052188179294, 3.50561592638930415211628589898, 4.27516766227616299875620750773, 4.94946932056961628938788272653, 5.34465390238234539769941603269, 6.08086898664347684779911898670, 6.35093675030024605547797105182, 7.21011486025012968461358392795, 7.61178950661489259753692982695, 7.949835094309691188515700603833, 8.393621315860024309714270669955, 8.912382805672646779752354853771, 9.037207096057459242685360680499, 9.833323138282610486409897849216, 9.909433120286834848958008835898, 10.57712839766422400962204337484, 10.76478819585551596315768989104

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