Properties

Label 4-7644e2-1.1-c1e2-0-7
Degree $4$
Conductor $58430736$
Sign $1$
Analytic cond. $3725.59$
Root an. cond. $7.81265$
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  − 2·3-s − 5-s + 3·9-s + 4·11-s + 2·13-s + 2·15-s − 7·19-s + 3·23-s − 25-s − 4·27-s − 3·29-s + 31-s − 8·33-s + 2·37-s − 4·39-s − 6·41-s + 5·43-s − 3·45-s + 7·47-s + 5·53-s − 4·55-s + 14·57-s + 16·59-s − 12·61-s − 2·65-s + 14·67-s − 6·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.60·19-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.557·29-s + 0.179·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.762·43-s − 0.447·45-s + 1.02·47-s + 0.686·53-s − 0.539·55-s + 1.85·57-s + 2.08·59-s − 1.53·61-s − 0.248·65-s + 1.71·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(58430736\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3725.59\)
Root analytic conductor: \(7.81265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 58430736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.894418796\)
\(L(\frac12)\) \(\approx\) \(1.894418796\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T + 98 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 190 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 29 T + 396 T^{2} + 29 p T^{3} + p^{2} T^{4} \)
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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

−7.83422983549892817206374131692, −7.80646403650249034745819303763, −7.17339974202224055847078632513, −6.91374354999532100307991331989, −6.58621968710743022131789556477, −6.45196946005752336279303381949, −5.83439717668421762673464716387, −5.80124821209625027960061926441, −5.25697696782888812571226921188, −4.93447468015548704998185027680, −4.33948800390894330558888376112, −4.23794332031473584731802141805, −3.74456166261019688731733419535, −3.67961576420719775120124934790, −2.87873491698267737217202065508, −2.45659779218722649991064154283, −1.79461396008775756194560689391, −1.54989730858972343540937227116, −0.76092106459037558910641398196, −0.50785677616995654619015389337, 0.50785677616995654619015389337, 0.76092106459037558910641398196, 1.54989730858972343540937227116, 1.79461396008775756194560689391, 2.45659779218722649991064154283, 2.87873491698267737217202065508, 3.67961576420719775120124934790, 3.74456166261019688731733419535, 4.23794332031473584731802141805, 4.33948800390894330558888376112, 4.93447468015548704998185027680, 5.25697696782888812571226921188, 5.80124821209625027960061926441, 5.83439717668421762673464716387, 6.45196946005752336279303381949, 6.58621968710743022131789556477, 6.91374354999532100307991331989, 7.17339974202224055847078632513, 7.80646403650249034745819303763, 7.83422983549892817206374131692

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