Properties

Label 4-1632e2-1.1-c1e2-0-1
Degree $4$
Conductor $2663424$
Sign $1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 2·17-s − 8·19-s − 2·25-s − 4·41-s − 8·43-s − 2·49-s − 8·59-s + 8·67-s + 4·73-s + 81-s − 24·83-s + 12·89-s + 4·97-s + 16·107-s + 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s − 0.485·17-s − 1.83·19-s − 2/5·25-s − 0.624·41-s − 1.21·43-s − 2/7·49-s − 1.04·59-s + 0.977·67-s + 0.468·73-s + 1/9·81-s − 2.63·83-s + 1.27·89-s + 0.406·97-s + 1.54·107-s + 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2663424} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.253394829\)
\(L(\frac12)\) \(\approx\) \(1.253394829\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−7.66643847783662331491727887735, −7.06694925286556935352027326222, −6.77600645740645616349403873756, −6.44235013404215990367300044482, −5.97933664252796158852075932824, −5.56669173386029527603454231932, −4.95891298489770699391384730184, −4.51072908441005746778926140583, −4.29069031972477030392473592384, −3.60512740650267412739964674531, −3.24555909857976407870127946051, −2.48784527810785707123696698847, −1.98833378335270627179565432086, −1.53318243450965361161045557035, −0.42052544310490231543126167757, 0.42052544310490231543126167757, 1.53318243450965361161045557035, 1.98833378335270627179565432086, 2.48784527810785707123696698847, 3.24555909857976407870127946051, 3.60512740650267412739964674531, 4.29069031972477030392473592384, 4.51072908441005746778926140583, 4.95891298489770699391384730184, 5.56669173386029527603454231932, 5.97933664252796158852075932824, 6.44235013404215990367300044482, 6.77600645740645616349403873756, 7.06694925286556935352027326222, 7.66643847783662331491727887735

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