Properties

Label 4-1400e2-1.1-c0e2-0-0
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
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 − 2·7-s − 8-s − 2·9-s − 2·14-s − 16-s − 2·18-s − 2·23-s − 2·46-s + 3·49-s + 2·56-s + 4·63-s + 64-s + 2·71-s + 2·72-s + 2·79-s + 3·81-s + 3·98-s + 2·112-s + 2·113-s − 121-s + 4·126-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2-s − 2·7-s − 8-s − 2·9-s − 2·14-s − 16-s − 2·18-s − 2·23-s − 2·46-s + 3·49-s + 2·56-s + 4·63-s + 64-s + 2·71-s + 2·72-s + 2·79-s + 3·81-s + 3·98-s + 2·112-s + 2·113-s − 121-s + 4·126-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5614550726\)
\(L(\frac12)\) \(\approx\) \(0.5614550726\)
\(L(1)\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−9.843954593408458694068303863593, −9.493651268341040971641532956191, −9.303160052920402604799326578851, −8.704452215552103367009069110378, −8.469937811031851222126383239222, −8.044919118323763129273374989878, −7.50665489231399357733332891726, −6.89467839796931541184747272690, −6.37055667514839173299920024060, −6.07680643003824312163113423068, −6.06148820894904689528403539727, −5.29672123352404281747730163147, −5.20451022881704267442759786319, −4.33499180401528034990351055655, −3.85084614058781189252424500176, −3.40098446496291043402696903894, −3.24363039444552861579197778460, −2.37007642889111239523378475339, −2.36349393864433788223460608726, −0.49418645872110009508556534167, 0.49418645872110009508556534167, 2.36349393864433788223460608726, 2.37007642889111239523378475339, 3.24363039444552861579197778460, 3.40098446496291043402696903894, 3.85084614058781189252424500176, 4.33499180401528034990351055655, 5.20451022881704267442759786319, 5.29672123352404281747730163147, 6.06148820894904689528403539727, 6.07680643003824312163113423068, 6.37055667514839173299920024060, 6.89467839796931541184747272690, 7.50665489231399357733332891726, 8.044919118323763129273374989878, 8.469937811031851222126383239222, 8.704452215552103367009069110378, 9.303160052920402604799326578851, 9.493651268341040971641532956191, 9.843954593408458694068303863593

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