Properties

Label 4-13e4-1.1-c3e2-0-13
Degree $4$
Conductor $28561$
Sign $1$
Analytic cond. $99.4272$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·3-s − 4·4-s + 93·9-s + 56·12-s − 48·16-s − 54·17-s − 114·23-s − 58·25-s − 238·27-s − 138·29-s − 372·36-s + 170·43-s + 672·48-s − 179·49-s + 756·51-s + 852·53-s − 34·61-s + 448·64-s + 216·68-s + 1.59e3·69-s + 812·75-s − 2.48e3·79-s − 1.68e3·81-s + 1.93e3·87-s + 456·92-s + 232·100-s − 3.91e3·101-s + ⋯
L(s)  = 1  − 2.69·3-s − 1/2·4-s + 31/9·9-s + 1.34·12-s − 3/4·16-s − 0.770·17-s − 1.03·23-s − 0.463·25-s − 1.69·27-s − 0.883·29-s − 1.72·36-s + 0.602·43-s + 2.02·48-s − 0.521·49-s + 2.07·51-s + 2.20·53-s − 0.0713·61-s + 7/8·64-s + 0.385·68-s + 2.78·69-s + 1.25·75-s − 3.54·79-s − 2.31·81-s + 2.38·87-s + 0.516·92-s + 0.231·100-s − 3.85·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28561\)    =    \(13^{4}\)
Sign: $1$
Analytic conductor: \(99.4272\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{169} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 28561,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad13 \( 1 \)
good2$C_2^2$ \( 1 + p^{2} T^{2} + p^{6} T^{4} \)
3$C_2$ \( ( 1 + 7 T + p^{3} T^{2} )^{2} \)
5$C_2^2$ \( 1 + 58 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 179 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 2155 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 27 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 5915 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 57 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 69 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 54290 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 99719 T^{2} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 16745 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 85 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 90034 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 426 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 410395 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 + 17 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 574451 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 + 375115 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 231166 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 962026 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 1315951 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 300239 T^{2} + p^{6} T^{4} \)
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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

−12.19897748690645773708945667614, −11.48217719835871621332339167537, −11.06294483507506560520482804712, −10.96516330966958945477635777170, −10.08718045321594415059779411528, −9.921891229696085597481046122789, −9.057830829693235212687455397755, −8.581143343426175824590953335122, −7.80199539476799250738132440759, −6.83753034594576023534015011970, −6.81711797381145033805839784079, −5.94522954094110184298884003891, −5.58197403699315781284562515063, −5.20560348401395186663010199590, −4.23025508461146700014839836048, −4.19252532384516695831971535487, −2.54707575418626506781257122783, −1.29137390609072600850336473452, 0, 0, 1.29137390609072600850336473452, 2.54707575418626506781257122783, 4.19252532384516695831971535487, 4.23025508461146700014839836048, 5.20560348401395186663010199590, 5.58197403699315781284562515063, 5.94522954094110184298884003891, 6.81711797381145033805839784079, 6.83753034594576023534015011970, 7.80199539476799250738132440759, 8.581143343426175824590953335122, 9.057830829693235212687455397755, 9.921891229696085597481046122789, 10.08718045321594415059779411528, 10.96516330966958945477635777170, 11.06294483507506560520482804712, 11.48217719835871621332339167537, 12.19897748690645773708945667614

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