Properties

Label 4-8820e2-1.1-c1e2-0-2
Degree $4$
Conductor $77792400$
Sign $1$
Analytic cond. $4960.11$
Root an. cond. $8.39214$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·13-s + 14·19-s + 3·25-s + 12·29-s + 2·31-s − 2·37-s − 2·43-s − 12·47-s − 12·59-s + 8·61-s − 4·65-s − 2·67-s − 10·73-s + 22·79-s − 12·83-s − 12·89-s − 28·95-s − 16·97-s − 24·101-s + 26·103-s − 2·109-s + 36·113-s − 4·121-s − 4·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.554·13-s + 3.21·19-s + 3/5·25-s + 2.22·29-s + 0.359·31-s − 0.328·37-s − 0.304·43-s − 1.75·47-s − 1.56·59-s + 1.02·61-s − 0.496·65-s − 0.244·67-s − 1.17·73-s + 2.47·79-s − 1.31·83-s − 1.27·89-s − 2.87·95-s − 1.62·97-s − 2.38·101-s + 2.56·103-s − 0.191·109-s + 3.38·113-s − 0.363·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(77792400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4960.11\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8820} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 77792400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.543420924\)
\(L(\frac12)\) \(\approx\) \(3.543420924\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 117 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 153 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 196 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−7.80299411096270061986814118464, −7.75854169192619027655371942108, −7.11028074128686601255734189813, −7.08393657925861778611405906491, −6.50590356253095170190885934988, −6.43368215861921039141253124270, −5.67623828617640359292127385366, −5.60058961235877587276937059915, −5.06577986215996851387848864635, −4.86341816004139457962369503023, −4.35480373154367928825342415078, −4.18070680872129161695802889610, −3.45512994672663051644891800793, −3.25927010456485270974687211619, −2.96072215036706702734488614926, −2.75191700138546762487572789827, −1.72247674039738335076559472967, −1.53327493052458161839075221660, −0.798282995829753239027675648664, −0.60611729130721679839025420884, 0.60611729130721679839025420884, 0.798282995829753239027675648664, 1.53327493052458161839075221660, 1.72247674039738335076559472967, 2.75191700138546762487572789827, 2.96072215036706702734488614926, 3.25927010456485270974687211619, 3.45512994672663051644891800793, 4.18070680872129161695802889610, 4.35480373154367928825342415078, 4.86341816004139457962369503023, 5.06577986215996851387848864635, 5.60058961235877587276937059915, 5.67623828617640359292127385366, 6.43368215861921039141253124270, 6.50590356253095170190885934988, 7.08393657925861778611405906491, 7.11028074128686601255734189813, 7.75854169192619027655371942108, 7.80299411096270061986814118464

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