Properties

Label 4-6160e2-1.1-c1e2-0-10
Degree $4$
Conductor $37945600$
Sign $1$
Analytic cond. $2419.44$
Root an. cond. $7.01340$
Motivic weight $1$
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  + 2·5-s − 2·7-s − 4·9-s + 2·11-s − 4·13-s − 4·17-s + 4·23-s + 3·25-s − 12·29-s + 12·31-s − 4·35-s + 4·43-s − 8·45-s + 8·47-s + 3·49-s + 8·53-s + 4·55-s − 12·59-s − 16·61-s + 8·63-s − 8·65-s + 4·67-s − 8·71-s − 12·73-s − 4·77-s − 20·79-s + 7·81-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 4/3·9-s + 0.603·11-s − 1.10·13-s − 0.970·17-s + 0.834·23-s + 3/5·25-s − 2.22·29-s + 2.15·31-s − 0.676·35-s + 0.609·43-s − 1.19·45-s + 1.16·47-s + 3/7·49-s + 1.09·53-s + 0.539·55-s − 1.56·59-s − 2.04·61-s + 1.00·63-s − 0.992·65-s + 0.488·67-s − 0.949·71-s − 1.40·73-s − 0.455·77-s − 2.25·79-s + 7/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(37945600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2419.44\)
Root analytic conductor: \(7.01340\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 37945600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 92 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 164 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 250 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.68688458720070061845835381524, −7.49327368956507461160020217530, −7.15358351546254101976085321244, −6.88411464538576420339306210799, −6.20998939761085791182204724002, −6.20664932831447427557819146355, −5.71873819111444741285944103849, −5.64250746522713364796212185125, −4.96477846896915680601718248085, −4.71453486660785567636391838404, −4.17582616753130764743716611405, −3.97639587985772572970906002800, −3.13014187461231133808357420609, −2.97704764759380362073760594718, −2.52275961003715365304911276551, −2.35087845947482301143616888194, −1.52916152982208687347970503711, −1.15803678823190508109324474005, 0, 0, 1.15803678823190508109324474005, 1.52916152982208687347970503711, 2.35087845947482301143616888194, 2.52275961003715365304911276551, 2.97704764759380362073760594718, 3.13014187461231133808357420609, 3.97639587985772572970906002800, 4.17582616753130764743716611405, 4.71453486660785567636391838404, 4.96477846896915680601718248085, 5.64250746522713364796212185125, 5.71873819111444741285944103849, 6.20664932831447427557819146355, 6.20998939761085791182204724002, 6.88411464538576420339306210799, 7.15358351546254101976085321244, 7.49327368956507461160020217530, 7.68688458720070061845835381524

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