Properties

Label 4-8085e2-1.1-c1e2-0-3
Degree $4$
Conductor $65367225$
Sign $1$
Analytic cond. $4167.87$
Root an. cond. $8.03486$
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  − 2·2-s + 2·3-s + 4-s + 2·5-s − 4·6-s + 3·9-s − 4·10-s − 2·11-s + 2·12-s + 4·15-s + 16-s + 8·17-s − 6·18-s + 8·19-s + 2·20-s + 4·22-s − 8·23-s + 3·25-s + 4·27-s − 4·29-s − 8·30-s + 2·32-s − 4·33-s − 16·34-s + 3·36-s + 12·37-s − 16·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s + 9-s − 1.26·10-s − 0.603·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s + 1.94·17-s − 1.41·18-s + 1.83·19-s + 0.447·20-s + 0.852·22-s − 1.66·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s − 1.46·30-s + 0.353·32-s − 0.696·33-s − 2.74·34-s + 1/2·36-s + 1.97·37-s − 2.59·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65367225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4167.87\)
Root analytic conductor: \(8.03486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8085} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65367225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.119162487\)
\(L(\frac12)\) \(\approx\) \(3.119162487\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−7.962804422537913427106966580640, −7.925419787888793149562641091416, −7.50710586594100702950928824924, −7.30165974489213326592700589986, −6.70202027973124963999278136979, −6.37275172729579456124721431034, −6.00389369797315729981346851207, −5.45060156648847116531990534536, −5.29798926517950019313485618213, −5.12087331134951538378229128877, −4.30638547733632780092568033925, −4.00066331106684838849716205573, −3.46277127068768429339540672212, −3.22625515012726526816551293027, −2.82536210325336880229873980945, −2.34644151540959607782657199352, −1.89866558164410031499953653343, −1.52509007740516277347493077238, −0.808065135520453500186850518396, −0.67295794320558912994865498889, 0.67295794320558912994865498889, 0.808065135520453500186850518396, 1.52509007740516277347493077238, 1.89866558164410031499953653343, 2.34644151540959607782657199352, 2.82536210325336880229873980945, 3.22625515012726526816551293027, 3.46277127068768429339540672212, 4.00066331106684838849716205573, 4.30638547733632780092568033925, 5.12087331134951538378229128877, 5.29798926517950019313485618213, 5.45060156648847116531990534536, 6.00389369797315729981346851207, 6.37275172729579456124721431034, 6.70202027973124963999278136979, 7.30165974489213326592700589986, 7.50710586594100702950928824924, 7.925419787888793149562641091416, 7.962804422537913427106966580640

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