Properties

Label 4-103e2-1.1-c0e2-0-0
Degree $4$
Conductor $10609$
Sign $1$
Analytic cond. $0.00264233$
Root an. cond. $0.226723$
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 − 7-s + 2·9-s − 13-s + 14-s − 17-s − 2·18-s − 19-s − 23-s + 2·25-s + 26-s − 29-s + 32-s + 34-s + 38-s − 41-s + 46-s − 2·50-s + 58-s − 59-s − 61-s − 2·63-s − 64-s − 79-s + 3·81-s + 82-s − 83-s + ⋯
L(s)  = 1  − 2-s − 7-s + 2·9-s − 13-s + 14-s − 17-s − 2·18-s − 19-s − 23-s + 2·25-s + 26-s − 29-s + 32-s + 34-s + 38-s − 41-s + 46-s − 2·50-s + 58-s − 59-s − 61-s − 2·63-s − 64-s − 79-s + 3·81-s + 82-s − 83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10609\)    =    \(103^{2}\)
Sign: $1$
Analytic conductor: \(0.00264233\)
Root analytic conductor: \(0.226723\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{103} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10609,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1898228829\)
\(L(\frac12)\) \(\approx\) \(0.1898228829\)
\(L(1)\) 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)$
bad103$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
17$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
23$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
29$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
83$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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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

−14.35488539200342705454252134633, −13.68380576693065448866038657735, −13.06999783634067273401280863882, −12.70894104994314844385392810633, −12.51791807552834908624905200911, −11.72604188680177212338039429439, −10.91161638857600902781983446815, −10.34214505526155332385890590804, −9.959707971151309180493693391976, −9.587760247432759093520355809873, −8.974746013144302001889982996005, −8.593356723596807581237004373057, −7.69809741917201771363481039022, −7.07266036729859331897519459089, −6.71017783955903441438614047649, −6.03454633619783779298064033402, −4.64753948580894546247106478607, −4.46895357445124374556423870030, −3.28396214040166531134869327212, −2.00946183109297238095017317494, 2.00946183109297238095017317494, 3.28396214040166531134869327212, 4.46895357445124374556423870030, 4.64753948580894546247106478607, 6.03454633619783779298064033402, 6.71017783955903441438614047649, 7.07266036729859331897519459089, 7.69809741917201771363481039022, 8.593356723596807581237004373057, 8.974746013144302001889982996005, 9.587760247432759093520355809873, 9.959707971151309180493693391976, 10.34214505526155332385890590804, 10.91161638857600902781983446815, 11.72604188680177212338039429439, 12.51791807552834908624905200911, 12.70894104994314844385392810633, 13.06999783634067273401280863882, 13.68380576693065448866038657735, 14.35488539200342705454252134633

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