Properties

Label 4-79e2-1.1-c0e2-0-0
Degree $4$
Conductor $6241$
Sign $1$
Analytic cond. $0.00155442$
Root an. cond. $0.198560$
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-s − 5-s + 2·9-s + 10-s − 11-s − 13-s − 2·18-s − 19-s + 22-s − 23-s + 26-s − 31-s + 32-s + 38-s − 2·45-s + 46-s + 2·49-s + 55-s + 62-s − 64-s + 65-s − 67-s − 73-s + 2·79-s + 3·81-s + 4·83-s − 89-s + ⋯
L(s)  = 1  − 2-s − 5-s + 2·9-s + 10-s − 11-s − 13-s − 2·18-s − 19-s + 22-s − 23-s + 26-s − 31-s + 32-s + 38-s − 2·45-s + 46-s + 2·49-s + 55-s + 62-s − 64-s + 65-s − 67-s − 73-s + 2·79-s + 3·81-s + 4·83-s − 89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6241\)    =    \(79^{2}\)
Sign: $1$
Analytic conductor: \(0.00155442\)
Root analytic conductor: \(0.198560\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{79} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6241,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1523062674\)
\(L(\frac12)\) \(\approx\) \(0.1523062674\)
\(L(1)\) 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)$
bad79$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
13$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
23$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
97$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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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

−15.09438089440591022273049327848, −14.79301903017583787602292860928, −13.55274723202347412692925946328, −13.49476825961692915439080389739, −12.47946133071320119957371003723, −12.41280998218477785723439677217, −11.81774686271880974772720205144, −10.84468432527915440022069005515, −10.41921929706756148110282082287, −10.00582281033173329485446747233, −9.369771806913826376759339060543, −8.873978905013962342505313744455, −7.87413839665017329527808259336, −7.80057654256192655708865026236, −7.13654942825154090841373972766, −6.38007356073625378647747108423, −5.19349650437579468072287387020, −4.39037048839329186211077735661, −3.84845369197390074372926468934, −2.25042492444233676394569070777, 2.25042492444233676394569070777, 3.84845369197390074372926468934, 4.39037048839329186211077735661, 5.19349650437579468072287387020, 6.38007356073625378647747108423, 7.13654942825154090841373972766, 7.80057654256192655708865026236, 7.87413839665017329527808259336, 8.873978905013962342505313744455, 9.369771806913826376759339060543, 10.00582281033173329485446747233, 10.41921929706756148110282082287, 10.84468432527915440022069005515, 11.81774686271880974772720205144, 12.41280998218477785723439677217, 12.47946133071320119957371003723, 13.49476825961692915439080389739, 13.55274723202347412692925946328, 14.79301903017583787602292860928, 15.09438089440591022273049327848

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