Properties

Label 4-2520e2-1.1-c1e2-0-9
Degree $4$
Conductor $6350400$
Sign $1$
Analytic cond. $404.907$
Root an. cond. $4.48578$
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  − 5-s − 4·7-s + 3·11-s + 2·13-s + 7·19-s − 5·23-s − 6·31-s + 4·35-s − 3·37-s + 6·41-s + 16·43-s − 47-s + 9·49-s + 5·53-s − 3·55-s − 4·59-s + 8·61-s − 2·65-s + 12·71-s + 14·73-s − 12·77-s + 16·79-s + 32·83-s + 6·89-s − 8·91-s − 7·95-s + 32·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.904·11-s + 0.554·13-s + 1.60·19-s − 1.04·23-s − 1.07·31-s + 0.676·35-s − 0.493·37-s + 0.937·41-s + 2.43·43-s − 0.145·47-s + 9/7·49-s + 0.686·53-s − 0.404·55-s − 0.520·59-s + 1.02·61-s − 0.248·65-s + 1.42·71-s + 1.63·73-s − 1.36·77-s + 1.80·79-s + 3.51·83-s + 0.635·89-s − 0.838·91-s − 0.718·95-s + 3.24·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6350400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(404.907\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6350400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.348209209\)
\(L(\frac12)\) \(\approx\) \(2.348209209\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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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

−9.260265202689458243795270816822, −8.902864024088460803182879394994, −8.301940240250920807783633374213, −7.891368604562447538540632708795, −7.47426403077949028117787254963, −7.33266499804856655973525924971, −6.71230022867760360007390635906, −6.43816277742489665642610738531, −5.93250451670301538326087697393, −5.88581021447571321201622606447, −5.14080045295102106053661375082, −4.85570130901002677467830035324, −3.96807218411655335046076459677, −3.89184152053049030664057411143, −3.36831930887904137204450785417, −3.28250322732277928044848727034, −2.17761410624789828117848272655, −2.17116471879093239034055719207, −0.831904075491864375468347720537, −0.74055757516995056951015704646, 0.74055757516995056951015704646, 0.831904075491864375468347720537, 2.17116471879093239034055719207, 2.17761410624789828117848272655, 3.28250322732277928044848727034, 3.36831930887904137204450785417, 3.89184152053049030664057411143, 3.96807218411655335046076459677, 4.85570130901002677467830035324, 5.14080045295102106053661375082, 5.88581021447571321201622606447, 5.93250451670301538326087697393, 6.43816277742489665642610738531, 6.71230022867760360007390635906, 7.33266499804856655973525924971, 7.47426403077949028117787254963, 7.891368604562447538540632708795, 8.301940240250920807783633374213, 8.902864024088460803182879394994, 9.260265202689458243795270816822

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