Properties

Label 4-35e4-1.1-c1e2-0-13
Degree $4$
Conductor $1500625$
Sign $1$
Analytic cond. $95.6811$
Root an. cond. $3.12756$
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 − 2·4-s − 9-s − 6·11-s + 4·12-s − 6·13-s + 2·17-s + 12·19-s − 12·23-s + 6·27-s − 6·29-s + 12·31-s + 12·33-s + 2·36-s + 4·37-s + 12·39-s − 4·41-s − 4·43-s + 12·44-s + 6·47-s − 4·51-s + 12·52-s − 24·57-s + 4·59-s + 8·64-s + 8·67-s − 4·68-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1/3·9-s − 1.80·11-s + 1.15·12-s − 1.66·13-s + 0.485·17-s + 2.75·19-s − 2.50·23-s + 1.15·27-s − 1.11·29-s + 2.15·31-s + 2.08·33-s + 1/3·36-s + 0.657·37-s + 1.92·39-s − 0.624·41-s − 0.609·43-s + 1.80·44-s + 0.875·47-s − 0.560·51-s + 1.66·52-s − 3.17·57-s + 0.520·59-s + 64-s + 0.977·67-s − 0.485·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(95.6811\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1500625,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 18 T + 257 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
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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

−9.727562904682335687954883051916, −9.404704391189193060787404201610, −8.476890831530819554820204611247, −8.351562204940317397312430777185, −7.74466456519532638811131211242, −7.68817369756227385507554632767, −7.07874004748954413403443473732, −6.61232409023256412721584540770, −5.71512687113582828835604201501, −5.68148645017007703975725909143, −5.32438407152611259801184244192, −5.06592918906423014479992161523, −4.40556925606388490498667873035, −4.10883174836056285811580572450, −3.14801619989417113147300539746, −2.78840361055997743144197075966, −2.28171992430185593804772650813, −1.12557278961490605786276875128, 0, 0, 1.12557278961490605786276875128, 2.28171992430185593804772650813, 2.78840361055997743144197075966, 3.14801619989417113147300539746, 4.10883174836056285811580572450, 4.40556925606388490498667873035, 5.06592918906423014479992161523, 5.32438407152611259801184244192, 5.68148645017007703975725909143, 5.71512687113582828835604201501, 6.61232409023256412721584540770, 7.07874004748954413403443473732, 7.68817369756227385507554632767, 7.74466456519532638811131211242, 8.351562204940317397312430777185, 8.476890831530819554820204611247, 9.404704391189193060787404201610, 9.727562904682335687954883051916

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