Properties

Label 4-260e2-1.1-c1e2-0-6
Degree $4$
Conductor $67600$
Sign $1$
Analytic cond. $4.31023$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4·5-s + 8·7-s + 8·9-s − 8·11-s + 4·13-s − 16·15-s − 2·17-s − 32·21-s + 8·23-s + 11·25-s − 12·27-s − 4·31-s + 32·33-s + 32·35-s + 4·37-s − 16·39-s − 14·41-s − 4·43-s + 32·45-s + 16·47-s + 34·49-s + 8·51-s + 2·53-s − 32·55-s + 16·61-s + 64·63-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s + 3.02·7-s + 8/3·9-s − 2.41·11-s + 1.10·13-s − 4.13·15-s − 0.485·17-s − 6.98·21-s + 1.66·23-s + 11/5·25-s − 2.30·27-s − 0.718·31-s + 5.57·33-s + 5.40·35-s + 0.657·37-s − 2.56·39-s − 2.18·41-s − 0.609·43-s + 4.77·45-s + 2.33·47-s + 34/7·49-s + 1.12·51-s + 0.274·53-s − 4.31·55-s + 2.04·61-s + 8.06·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(67600\)    =    \(2^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.31023\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 67600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198429601\)
\(L(\frac12)\) \(\approx\) \(1.198429601\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91572947736811339150682310475, −11.56001188497679850968467800127, −11.06807165133341031682847075155, −10.97813881202735514051432259865, −10.43787775355233584242307940189, −10.43322287668835209076996298853, −9.556875962139852685408823925902, −8.560489148006415448741954079110, −8.546805655811458275275887243482, −7.78562916221885131163916155779, −7.11899985644777035282381386491, −6.66718588273627958576295072363, −5.63919735900342345409413247675, −5.62196837545831953575241786447, −5.13376448460541982066543075981, −5.05745135488930253863332377076, −4.29914916645939360221526187171, −2.59829359078786913488957600660, −1.84108772581657935865367274574, −1.11894348034076687966743047826, 1.11894348034076687966743047826, 1.84108772581657935865367274574, 2.59829359078786913488957600660, 4.29914916645939360221526187171, 5.05745135488930253863332377076, 5.13376448460541982066543075981, 5.62196837545831953575241786447, 5.63919735900342345409413247675, 6.66718588273627958576295072363, 7.11899985644777035282381386491, 7.78562916221885131163916155779, 8.546805655811458275275887243482, 8.560489148006415448741954079110, 9.556875962139852685408823925902, 10.43322287668835209076996298853, 10.43787775355233584242307940189, 10.97813881202735514051432259865, 11.06807165133341031682847075155, 11.56001188497679850968467800127, 11.91572947736811339150682310475

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