Properties

Label 4-260e2-1.1-c1e2-0-5
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  + 2·2-s + 2·4-s − 4·5-s − 8·10-s + 4·13-s − 4·16-s + 10·17-s − 8·20-s + 11·25-s + 8·26-s − 8·32-s + 20·34-s + 14·37-s + 22·50-s + 8·52-s + 10·53-s − 24·61-s − 8·64-s − 16·65-s + 20·68-s + 22·73-s + 28·74-s + 16·80-s − 9·81-s − 40·85-s + 20·89-s + 26·97-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.78·5-s − 2.52·10-s + 1.10·13-s − 16-s + 2.42·17-s − 1.78·20-s + 11/5·25-s + 1.56·26-s − 1.41·32-s + 3.42·34-s + 2.30·37-s + 3.11·50-s + 1.10·52-s + 1.37·53-s − 3.07·61-s − 64-s − 1.98·65-s + 2.42·68-s + 2.57·73-s + 3.25·74-s + 1.78·80-s − 81-s − 4.33·85-s + 2.11·89-s + 2.63·97-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

\(L(1)\) \(\approx\) \(2.281104873\)
\(L(\frac12)\) \(\approx\) \(2.281104873\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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

−12.19808920026082555490503275323, −11.83479989647058711874851966046, −11.62520358704486164764320485690, −10.94764304595673512092141508531, −10.67440659049052817001036729708, −9.985301004627063300813341812259, −9.197057706338527608000351424781, −8.893088393591458338606294055737, −7.956562089194112718817643091403, −7.83112531043615601525828886727, −7.46251951013669094485232831439, −6.37087300067550339343483220817, −6.33534823329739839119619728991, −5.37054128530512854989016519804, −5.06211686255983221878751417400, −4.04717583880578270689560921770, −3.99713765636240581374531385895, −3.27607928396725695438596259054, −2.78410593719181404868377758581, −1.05784920675631122175991650457, 1.05784920675631122175991650457, 2.78410593719181404868377758581, 3.27607928396725695438596259054, 3.99713765636240581374531385895, 4.04717583880578270689560921770, 5.06211686255983221878751417400, 5.37054128530512854989016519804, 6.33534823329739839119619728991, 6.37087300067550339343483220817, 7.46251951013669094485232831439, 7.83112531043615601525828886727, 7.956562089194112718817643091403, 8.893088393591458338606294055737, 9.197057706338527608000351424781, 9.985301004627063300813341812259, 10.67440659049052817001036729708, 10.94764304595673512092141508531, 11.62520358704486164764320485690, 11.83479989647058711874851966046, 12.19808920026082555490503275323

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