Properties

Label 4-495e2-1.1-c0e2-0-3
Degree $4$
Conductor $245025$
Sign $1$
Analytic cond. $0.0610273$
Root an. cond. $0.497028$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·11-s − 16-s + 3·25-s − 4·55-s − 2·80-s − 4·89-s + 3·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2·5-s − 2·11-s − 16-s + 3·25-s − 4·55-s − 2·80-s − 4·89-s + 3·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(245025\)    =    \(3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.0610273\)
Root analytic conductor: \(0.497028\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{495} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 245025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9280014355\)
\(L(\frac12)\) \(\approx\) \(0.9280014355\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−11.20527720293515460316132033344, −10.76660777022994655235228169128, −10.47715785806306955454097860720, −10.08336258337823404344411885192, −9.563289442620246560947755463579, −9.413808162322701135677632800148, −8.674904304332766192542308775724, −8.401864268159942187199920064110, −7.83609449669001126086671876843, −7.02468406012674087089570360911, −6.96188246317849590672782469095, −6.18230644516051652951258386661, −5.68195058062788999199735320572, −5.45654632625930193207409868945, −4.82582518091095724664559359195, −4.46091705236518078993507275027, −3.32019965927162093848879516456, −2.56324593771758606772819653651, −2.40520655674812140903479631954, −1.51719217207667209008065103625, 1.51719217207667209008065103625, 2.40520655674812140903479631954, 2.56324593771758606772819653651, 3.32019965927162093848879516456, 4.46091705236518078993507275027, 4.82582518091095724664559359195, 5.45654632625930193207409868945, 5.68195058062788999199735320572, 6.18230644516051652951258386661, 6.96188246317849590672782469095, 7.02468406012674087089570360911, 7.83609449669001126086671876843, 8.401864268159942187199920064110, 8.674904304332766192542308775724, 9.413808162322701135677632800148, 9.563289442620246560947755463579, 10.08336258337823404344411885192, 10.47715785806306955454097860720, 10.76660777022994655235228169128, 11.20527720293515460316132033344

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