Properties

Label 4-260e2-1.1-c1e2-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 7-s + 3·9-s − 3·11-s + 2·13-s + 2·15-s + 3·17-s − 5·19-s − 21-s − 9·23-s + 3·25-s − 8·27-s + 9·29-s + 16·31-s + 3·33-s − 2·35-s + 7·37-s − 2·39-s − 3·41-s + 43-s − 6·45-s + 7·49-s − 3·51-s + 12·53-s + 6·55-s + 5·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s + 9-s − 0.904·11-s + 0.554·13-s + 0.516·15-s + 0.727·17-s − 1.14·19-s − 0.218·21-s − 1.87·23-s + 3/5·25-s − 1.53·27-s + 1.67·29-s + 2.87·31-s + 0.522·33-s − 0.338·35-s + 1.15·37-s − 0.320·39-s − 0.468·41-s + 0.152·43-s − 0.894·45-s + 49-s − 0.420·51-s + 1.64·53-s + 0.809·55-s + 0.662·57-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: induced by $\chi_{260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 67600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.005156350\)
\(L(\frac12)\) \(\approx\) \(1.005156350\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−12.00492380768242560403405660627, −11.98305682037253463131741967335, −11.35753933106026292011482398084, −10.73252545284597420117471941249, −10.30509448922771641790855168684, −10.12987672505612879469113749589, −9.545910646661125405353337508357, −8.547322484348089346348842264120, −8.282733700546634353088809613062, −7.907038550521635541879595245296, −7.42147685506373683673133857360, −6.72667613616255717497304742582, −6.12218659300924945656328048533, −5.80750831831955318944786424904, −4.83472529881436202340352022996, −4.37497526960047177461463069333, −4.05170959812265054789947377536, −3.03568569313595587159094219180, −2.15027365314411262605981248312, −0.856557334826550739761263451070, 0.856557334826550739761263451070, 2.15027365314411262605981248312, 3.03568569313595587159094219180, 4.05170959812265054789947377536, 4.37497526960047177461463069333, 4.83472529881436202340352022996, 5.80750831831955318944786424904, 6.12218659300924945656328048533, 6.72667613616255717497304742582, 7.42147685506373683673133857360, 7.907038550521635541879595245296, 8.282733700546634353088809613062, 8.547322484348089346348842264120, 9.545910646661125405353337508357, 10.12987672505612879469113749589, 10.30509448922771641790855168684, 10.73252545284597420117471941249, 11.35753933106026292011482398084, 11.98305682037253463131741967335, 12.00492380768242560403405660627

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