Properties

Label 4-5733e2-1.1-c1e2-0-0
Degree $4$
Conductor $32867289$
Sign $1$
Analytic cond. $2095.64$
Root an. cond. $6.76596$
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  + 3·2-s + 4·4-s + 3·8-s + 6·11-s + 2·13-s + 3·16-s − 6·17-s − 6·19-s + 18·22-s + 12·23-s − 5·25-s + 6·26-s − 10·31-s + 6·32-s − 18·34-s − 4·37-s − 18·38-s − 16·43-s + 24·44-s + 36·46-s − 6·47-s − 15·50-s + 8·52-s + 6·53-s + 6·59-s − 6·61-s − 30·62-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.06·8-s + 1.80·11-s + 0.554·13-s + 3/4·16-s − 1.45·17-s − 1.37·19-s + 3.83·22-s + 2.50·23-s − 25-s + 1.17·26-s − 1.79·31-s + 1.06·32-s − 3.08·34-s − 0.657·37-s − 2.91·38-s − 2.43·43-s + 3.61·44-s + 5.30·46-s − 0.875·47-s − 2.12·50-s + 1.10·52-s + 0.824·53-s + 0.781·59-s − 0.768·61-s − 3.81·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32867289\)    =    \(3^{4} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2095.64\)
Root analytic conductor: \(6.76596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5733} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32867289,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.778127144\)
\(L(\frac12)\) \(\approx\) \(7.778127144\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 173 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−8.369299252842710208175005943255, −7.83846641756300875643215997581, −7.35569642603253922834594100267, −6.82027312413841126972296451808, −6.71447293615684000796789587535, −6.62110409224774064291736982500, −6.00585498162097442300494898944, −5.74054550027058640046509304207, −5.31309501498823282379640591745, −4.95943044603200181664775183734, −4.51840173910174361416488321935, −4.38639452861344996783464811830, −3.85660452659998974759337628235, −3.70988475403844324422091877518, −3.12205114265994011193916987139, −3.07314331307448128331716543189, −2.08395551298999192664952506648, −1.81049481444218874912491553974, −1.41447437409073957235408291139, −0.47441756873978943461784791744, 0.47441756873978943461784791744, 1.41447437409073957235408291139, 1.81049481444218874912491553974, 2.08395551298999192664952506648, 3.07314331307448128331716543189, 3.12205114265994011193916987139, 3.70988475403844324422091877518, 3.85660452659998974759337628235, 4.38639452861344996783464811830, 4.51840173910174361416488321935, 4.95943044603200181664775183734, 5.31309501498823282379640591745, 5.74054550027058640046509304207, 6.00585498162097442300494898944, 6.62110409224774064291736982500, 6.71447293615684000796789587535, 6.82027312413841126972296451808, 7.35569642603253922834594100267, 7.83846641756300875643215997581, 8.369299252842710208175005943255

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