Properties

Label 4-370e2-1.1-c1e2-0-11
Degree $4$
Conductor $136900$
Sign $1$
Analytic cond. $8.72886$
Root an. cond. $1.71885$
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 − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 6·7-s − 4·8-s − 4·10-s − 4·11-s − 6·12-s − 4·13-s + 12·14-s − 4·15-s + 5·16-s + 4·17-s + 2·19-s + 6·20-s + 12·21-s + 8·22-s − 16·23-s + 8·24-s + 3·25-s + 8·26-s + 2·27-s − 18·28-s − 4·29-s + 8·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 2.26·7-s − 1.41·8-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 1.10·13-s + 3.20·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 0.458·19-s + 1.34·20-s + 2.61·21-s + 1.70·22-s − 3.33·23-s + 1.63·24-s + 3/5·25-s + 1.56·26-s + 0.384·27-s − 3.40·28-s − 0.742·29-s + 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(136900\)    =    \(2^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(8.72886\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 136900,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 204 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 14 T + 212 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−10.96571294432696385833204859678, −10.46326174053223918931304521411, −10.10950998108117998920238357827, −9.806964005969077029406927965912, −9.470504733169353546690605268090, −9.380578784321491564422729633152, −8.157429517619222708056633769758, −8.038043063656137002117730924866, −7.45397717871850861724929250526, −6.77938727050772620060864557584, −6.26288012231895916120009879567, −6.13419658720380308842395948086, −5.38554572621642410432621906057, −5.33494467746224220922225621034, −3.92751481055437510594866452905, −3.13746761846583704719760510993, −2.61882371334183910411617060775, −1.81490457807024976965961734852, 0, 0, 1.81490457807024976965961734852, 2.61882371334183910411617060775, 3.13746761846583704719760510993, 3.92751481055437510594866452905, 5.33494467746224220922225621034, 5.38554572621642410432621906057, 6.13419658720380308842395948086, 6.26288012231895916120009879567, 6.77938727050772620060864557584, 7.45397717871850861724929250526, 8.038043063656137002117730924866, 8.157429517619222708056633769758, 9.380578784321491564422729633152, 9.470504733169353546690605268090, 9.806964005969077029406927965912, 10.10950998108117998920238357827, 10.46326174053223918931304521411, 10.96571294432696385833204859678

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