Properties

Label 4-370e2-1.1-c1e2-0-1
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s − 4·7-s + 16-s − 4·17-s + 4·19-s + 4·20-s + 11·25-s + 4·28-s − 14·29-s − 8·31-s + 16·35-s − 12·37-s − 4·47-s + 8·49-s − 2·53-s + 4·59-s + 2·61-s − 64-s + 4·68-s + 24·71-s − 10·73-s − 4·76-s + 24·79-s − 4·80-s − 9·81-s + 8·83-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1.51·7-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.755·28-s − 2.59·29-s − 1.43·31-s + 2.70·35-s − 1.97·37-s − 0.583·47-s + 8/7·49-s − 0.274·53-s + 0.520·59-s + 0.256·61-s − 1/8·64-s + 0.485·68-s + 2.84·71-s − 1.17·73-s − 0.458·76-s + 2.70·79-s − 0.447·80-s − 81-s + 0.878·83-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: induced by $\chi_{370} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 136900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2766054425\)
\(L(\frac12)\) \(\approx\) \(0.2766054425\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_2$ \( 1 + 12 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−11.52408095105280412824585195220, −11.05837547175172259104115780760, −11.01953573661147156632064910491, −10.25618407552202437619726152264, −9.623516144900915431462891346675, −9.258357554722968770592435204692, −9.050472398493849135696134044775, −8.226410853831396909943080683464, −8.049025178886690148935666509297, −7.22383955773552856424802437728, −6.99996654914562854651426053592, −6.68162990850165074635186615510, −5.52774474511993154997395398781, −5.48336745880939156782373721048, −4.52095980663305453796274492821, −3.94763464741997591106777175486, −3.38876272255592423234517035955, −3.31976316621932133557067001860, −1.99332978472701014131446014688, −0.34922906211715481459469904070, 0.34922906211715481459469904070, 1.99332978472701014131446014688, 3.31976316621932133557067001860, 3.38876272255592423234517035955, 3.94763464741997591106777175486, 4.52095980663305453796274492821, 5.48336745880939156782373721048, 5.52774474511993154997395398781, 6.68162990850165074635186615510, 6.99996654914562854651426053592, 7.22383955773552856424802437728, 8.049025178886690148935666509297, 8.226410853831396909943080683464, 9.050472398493849135696134044775, 9.258357554722968770592435204692, 9.623516144900915431462891346675, 10.25618407552202437619726152264, 11.01953573661147156632064910491, 11.05837547175172259104115780760, 11.52408095105280412824585195220

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