Properties

Label 4-7800e2-1.1-c1e2-0-1
Degree $4$
Conductor $60840000$
Sign $1$
Analytic cond. $3879.21$
Root an. cond. $7.89197$
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  − 2·3-s + 2·7-s + 3·9-s − 2·11-s − 2·13-s + 2·17-s − 4·19-s − 4·21-s + 4·23-s − 4·27-s − 6·29-s + 6·31-s + 4·33-s − 4·37-s + 4·39-s − 4·43-s + 10·47-s − 9·49-s − 4·51-s − 6·53-s + 8·57-s − 10·59-s + 2·61-s + 6·63-s + 10·67-s − 8·69-s − 4·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s − 0.769·27-s − 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.609·43-s + 1.45·47-s − 9/7·49-s − 0.560·51-s − 0.824·53-s + 1.05·57-s − 1.30·59-s + 0.256·61-s + 0.755·63-s + 1.22·67-s − 0.963·69-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60840000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3879.21\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 60840000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 69 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 141 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 141 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_4$ \( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−7.65505552484139893910982204867, −7.34546056978665090010019695468, −6.86364996755545557494333988767, −6.76148596586763014391602417703, −6.10534376852989647463977389456, −6.08531503458998691003173646863, −5.42986288125131486958830049550, −5.29349847439803882073004442338, −4.86700783069264527152267584865, −4.72456304988199802748126408139, −4.09941838692569730906743495863, −4.00923235246730581677128086819, −3.19958270232897897499669005635, −2.99848048223864324548280557079, −2.31899830698031171876791112009, −2.00770662598978189488872407011, −1.32737307812327504819314845613, −1.11390845387100822871916869345, 0, 0, 1.11390845387100822871916869345, 1.32737307812327504819314845613, 2.00770662598978189488872407011, 2.31899830698031171876791112009, 2.99848048223864324548280557079, 3.19958270232897897499669005635, 4.00923235246730581677128086819, 4.09941838692569730906743495863, 4.72456304988199802748126408139, 4.86700783069264527152267584865, 5.29349847439803882073004442338, 5.42986288125131486958830049550, 6.08531503458998691003173646863, 6.10534376852989647463977389456, 6.76148596586763014391602417703, 6.86364996755545557494333988767, 7.34546056978665090010019695468, 7.65505552484139893910982204867

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