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Properties

Label 4-117e2-1.1-c1e2-0-18
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 2·7-s − 4·8-s + 6·9-s + 8·10-s − 2·11-s − 6·12-s − 13-s + 4·14-s + 12·15-s + 8·16-s − 10·17-s − 12·18-s − 12·19-s − 8·20-s + 6·21-s + 4·22-s + 3·23-s + 12·24-s + 5·25-s + 2·26-s − 9·27-s − 4·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 0.755·7-s − 1.41·8-s + 2·9-s + 2.52·10-s − 0.603·11-s − 1.73·12-s − 0.277·13-s + 1.06·14-s + 3.09·15-s + 2·16-s − 2.42·17-s − 2.82·18-s − 2.75·19-s − 1.78·20-s + 1.30·21-s + 0.852·22-s + 0.625·23-s + 2.44·24-s + 25-s + 0.392·26-s − 1.73·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.872822\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
13$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−12.88404769380007274792023612784, −12.36886985559743262914532758057, −12.34412890230565534100741464708, −11.38179457796890927224920857107, −11.19543785121556848574998822614, −10.71820635543410582069665704574, −10.39039528355302913521713267168, −9.567657908037077911008199632307, −8.891939893687426618524028996333, −8.563269284622916452625217341821, −7.905208079325265537173997537277, −7.16409432509687099870075453860, −6.54890166747572833456200045986, −6.39603500228795107885277515346, −5.38590008515613444261790887688, −4.36470287146865125139193462229, −3.99955899920403288461230269357, −2.50964273036828341017701981849, 0, 0, 2.50964273036828341017701981849, 3.99955899920403288461230269357, 4.36470287146865125139193462229, 5.38590008515613444261790887688, 6.39603500228795107885277515346, 6.54890166747572833456200045986, 7.16409432509687099870075453860, 7.905208079325265537173997537277, 8.563269284622916452625217341821, 8.891939893687426618524028996333, 9.567657908037077911008199632307, 10.39039528355302913521713267168, 10.71820635543410582069665704574, 11.19543785121556848574998822614, 11.38179457796890927224920857107, 12.34412890230565534100741464708, 12.36886985559743262914532758057, 12.88404769380007274792023612784

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