Properties

Degree $4$
Conductor $200704$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 4·7-s + 3·9-s + 3·11-s − 3·15-s − 9·17-s + 7·19-s + 4·21-s + 15·23-s + 25-s + 8·27-s + 12·29-s − 5·31-s + 3·33-s − 12·35-s − 5·37-s − 9·45-s − 3·47-s + 9·49-s − 9·51-s − 9·53-s − 9·55-s + 7·57-s + 9·59-s − 15·61-s + 12·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1.51·7-s + 9-s + 0.904·11-s − 0.774·15-s − 2.18·17-s + 1.60·19-s + 0.872·21-s + 3.12·23-s + 1/5·25-s + 1.53·27-s + 2.22·29-s − 0.898·31-s + 0.522·33-s − 2.02·35-s − 0.821·37-s − 1.34·45-s − 0.437·47-s + 9/7·49-s − 1.26·51-s − 1.23·53-s − 1.21·55-s + 0.927·57-s + 1.17·59-s − 1.92·61-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33446\)
\(L(\frac12)\) \(\approx\) \(2.33446\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 21 T + 236 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−11.28239683393158912996302462933, −11.19324079712049036509797958602, −10.48035298536420936514595817813, −10.08823741864753979791064582967, −9.118184707601892226556638047640, −9.115964111464866245583053545328, −8.472037503251095161179955563837, −8.390638836177386874955471337784, −7.43874436308053135156227671026, −7.39249821190401119780341503246, −6.83895733298257019307095787038, −6.52861183224446142689591277630, −5.35586227041837510054439036943, −4.85175543567999249403828963436, −4.37956877878679899619220354721, −4.28851592292334302512677377763, −3.16721748703777296941477063233, −2.93473144650272399592128102412, −1.66711781056939195470525437197, −1.09531638981840601788368977620, 1.09531638981840601788368977620, 1.66711781056939195470525437197, 2.93473144650272399592128102412, 3.16721748703777296941477063233, 4.28851592292334302512677377763, 4.37956877878679899619220354721, 4.85175543567999249403828963436, 5.35586227041837510054439036943, 6.52861183224446142689591277630, 6.83895733298257019307095787038, 7.39249821190401119780341503246, 7.43874436308053135156227671026, 8.390638836177386874955471337784, 8.472037503251095161179955563837, 9.115964111464866245583053545328, 9.118184707601892226556638047640, 10.08823741864753979791064582967, 10.48035298536420936514595817813, 11.19324079712049036509797958602, 11.28239683393158912996302462933

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