Properties

Label 4-960e2-1.1-c2e2-0-2
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $684.246$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 12·7-s + 7·9-s − 32·13-s − 4·19-s − 48·21-s − 5·25-s + 8·27-s + 36·31-s + 32·37-s + 128·39-s + 32·43-s + 10·49-s + 16·57-s − 164·61-s + 84·63-s + 48·67-s − 148·73-s + 20·75-s − 276·79-s − 95·81-s − 384·91-s − 144·93-s − 332·97-s − 52·103-s − 76·109-s − 128·111-s + ⋯
L(s)  = 1  − 4/3·3-s + 12/7·7-s + 7/9·9-s − 2.46·13-s − 0.210·19-s − 2.28·21-s − 1/5·25-s + 8/27·27-s + 1.16·31-s + 0.864·37-s + 3.28·39-s + 0.744·43-s + 0.204·49-s + 0.280·57-s − 2.68·61-s + 4/3·63-s + 0.716·67-s − 2.02·73-s + 4/15·75-s − 3.49·79-s − 1.17·81-s − 4.21·91-s − 1.54·93-s − 3.42·97-s − 0.504·103-s − 0.697·109-s − 1.15·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(684.246\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6492775163\)
\(L(\frac12)\) \(\approx\) \(0.6492775163\)
\(L(2)\) 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$C_2$ \( 1 + 4 T + p^{2} T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
good7$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 222 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 558 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \)
29$C_2^2$ \( 1 - 702 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 558 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 1998 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 5598 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6942 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 5598 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 138 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4958 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 + 166 T + p^{2} T^{2} )^{2} \)
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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

−10.12801115097464981351042589267, −9.647372128490916177056163988783, −9.477230093787840282589283622605, −8.580149292440752593299018175974, −8.441782060796695905835553108148, −7.67910929315828346176679765769, −7.63957138570768393105015165377, −7.09512248957116142536995196827, −6.66990364275682517353426478945, −6.04742201197605889267125029810, −5.62391781855864904832379054048, −5.26097200966135114157771146720, −4.73333019269184443964324844352, −4.36979397075462679772016663981, −4.35289058879286594739913613427, −2.82205985169475438449777201734, −2.76107949072036508443570627110, −1.77046346568683217860950922347, −1.30583539631100481521339032542, −0.29129745499956941348431915741, 0.29129745499956941348431915741, 1.30583539631100481521339032542, 1.77046346568683217860950922347, 2.76107949072036508443570627110, 2.82205985169475438449777201734, 4.35289058879286594739913613427, 4.36979397075462679772016663981, 4.73333019269184443964324844352, 5.26097200966135114157771146720, 5.62391781855864904832379054048, 6.04742201197605889267125029810, 6.66990364275682517353426478945, 7.09512248957116142536995196827, 7.63957138570768393105015165377, 7.67910929315828346176679765769, 8.441782060796695905835553108148, 8.580149292440752593299018175974, 9.477230093787840282589283622605, 9.647372128490916177056163988783, 10.12801115097464981351042589267

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