Properties

Label 4-2205e2-1.1-c3e2-0-13
Degree $4$
Conductor $4862025$
Sign $1$
Analytic cond. $16925.8$
Root an. cond. $11.4061$
Motivic weight $3$
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  + 2·2-s − 5·4-s + 10·5-s − 12·8-s + 20·10-s + 16·11-s + 76·13-s − 11·16-s − 124·17-s + 96·19-s − 50·20-s + 32·22-s + 16·23-s + 75·25-s + 152·26-s − 188·29-s + 120·31-s − 122·32-s − 248·34-s − 132·37-s + 192·38-s − 120·40-s + 100·41-s − 536·43-s − 80·44-s + 32·46-s − 928·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 5/8·4-s + 0.894·5-s − 0.530·8-s + 0.632·10-s + 0.438·11-s + 1.62·13-s − 0.171·16-s − 1.76·17-s + 1.15·19-s − 0.559·20-s + 0.310·22-s + 0.145·23-s + 3/5·25-s + 1.14·26-s − 1.20·29-s + 0.695·31-s − 0.673·32-s − 1.25·34-s − 0.586·37-s + 0.819·38-s − 0.474·40-s + 0.380·41-s − 1.90·43-s − 0.274·44-s + 0.102·46-s − 2.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4862025\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16925.8\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4862025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 9 T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 16 T - 474 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 76 T + 5806 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 124 T + 13638 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 16 T + 15150 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 120 T + 60590 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 132 T + 70814 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 536 T + 200086 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 928 T + 408830 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 884 T + 460350 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 104 T + 80534 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 468 T + 494606 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1688 T + 1302310 T^{2} + 1688 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 136 T + 540446 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 508 T + 13078 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 432 T + 602142 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 584 T + 1172390 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1404 T + 1802390 T^{2} + 1404 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1188 T + 2161254 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.599265921864929036499316749946, −8.315036550518461727830497097497, −7.71609992171214037852700159109, −7.28240958691944810819915539234, −6.69553702158100015530847881002, −6.47899845466017870395267963637, −6.16253464100847283078368843895, −5.74022652167946432359543257661, −5.07177764738767782068616994677, −5.07151705439326347873258083741, −4.45300149853358879284929887087, −4.16144875295072540996771814574, −3.53177146455419447280546069377, −3.25500215829619149859317401992, −2.78969506184711206765446303603, −1.90198158430444244175174277216, −1.55110666680800339685094676219, −1.24357740107543545178297578724, 0, 0, 1.24357740107543545178297578724, 1.55110666680800339685094676219, 1.90198158430444244175174277216, 2.78969506184711206765446303603, 3.25500215829619149859317401992, 3.53177146455419447280546069377, 4.16144875295072540996771814574, 4.45300149853358879284929887087, 5.07151705439326347873258083741, 5.07177764738767782068616994677, 5.74022652167946432359543257661, 6.16253464100847283078368843895, 6.47899845466017870395267963637, 6.69553702158100015530847881002, 7.28240958691944810819915539234, 7.71609992171214037852700159109, 8.315036550518461727830497097497, 8.599265921864929036499316749946

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