Properties

Label 4-2268e2-1.1-c0e2-0-4
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $1.28115$
Root an. cond. $1.06389$
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  + 7-s − 3·13-s − 25-s + 2·37-s + 2·43-s − 3·61-s + 67-s + 79-s − 3·91-s + 3·97-s + 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 175-s + 179-s + 181-s + ⋯
L(s)  = 1  + 7-s − 3·13-s − 25-s + 2·37-s + 2·43-s − 3·61-s + 67-s + 79-s − 3·91-s + 3·97-s + 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 175-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.28115\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2268} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096798734\)
\(L(\frac12)\) \(\approx\) \(1.096798734\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
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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

−9.420056535421049446472641110578, −9.165248785724656983582651011530, −8.642851317633414069579636949229, −8.006807662205299319200684831470, −7.78153581652768383801341344239, −7.61235372126608453422466007365, −7.23799822220650440798619854488, −6.85433948495754649695117689604, −6.12118354656560318744628813082, −5.90758592807487326334319707482, −5.46934323291334221500532674465, −4.75202940456689191078574519267, −4.64567970474956593744742090617, −4.51977472756465596439625184697, −3.72391015436283279827454019357, −3.12270504489969288711812826615, −2.47442362313862663710996778273, −2.28481665526116421541602402999, −1.73068103862767334495426726424, −0.71197459327781752028770392355, 0.71197459327781752028770392355, 1.73068103862767334495426726424, 2.28481665526116421541602402999, 2.47442362313862663710996778273, 3.12270504489969288711812826615, 3.72391015436283279827454019357, 4.51977472756465596439625184697, 4.64567970474956593744742090617, 4.75202940456689191078574519267, 5.46934323291334221500532674465, 5.90758592807487326334319707482, 6.12118354656560318744628813082, 6.85433948495754649695117689604, 7.23799822220650440798619854488, 7.61235372126608453422466007365, 7.78153581652768383801341344239, 8.006807662205299319200684831470, 8.642851317633414069579636949229, 9.165248785724656983582651011530, 9.420056535421049446472641110578

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