Properties

Label 4-8820e2-1.1-c1e2-0-0
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

\(L(1)\) \(\approx\) \(3.592231719\)
\(L(\frac12)\) \(\approx\) \(3.592231719\)
\(L(\frac{3}{2})\) 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

−7.83105676710422415285207893752, −7.79076275214795887333765216091, −7.08241358759785289969996718264, −6.69934425875923768946988774096, −6.58880000276517616250572309187, −6.35075759102156449936861472647, −5.82611006553392563817109859347, −5.70386470058840429189856553935, −4.96171545632581238238031580895, −4.85698990207285509432897653804, −4.54348538671556260744623334951, −4.09161669522373807888201651537, −3.69011076642521755905085747470, −3.28421640942799455539292980423, −2.55068025912105479953401505325, −2.47455387140088110597940996929, −1.96612411066196486085001043291, −1.79209268752176196911526702585, −0.77929302375563234563813573220, −0.55688013296613931566906696318, 0.55688013296613931566906696318, 0.77929302375563234563813573220, 1.79209268752176196911526702585, 1.96612411066196486085001043291, 2.47455387140088110597940996929, 2.55068025912105479953401505325, 3.28421640942799455539292980423, 3.69011076642521755905085747470, 4.09161669522373807888201651537, 4.54348538671556260744623334951, 4.85698990207285509432897653804, 4.96171545632581238238031580895, 5.70386470058840429189856553935, 5.82611006553392563817109859347, 6.35075759102156449936861472647, 6.58880000276517616250572309187, 6.69934425875923768946988774096, 7.08241358759785289969996718264, 7.79076275214795887333765216091, 7.83105676710422415285207893752

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