Properties

Label 4-857500-1.1-c1e2-0-7
Degree $4$
Conductor $857500$
Sign $-1$
Analytic cond. $54.6749$
Root an. cond. $2.71923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 7-s + 4·8-s − 2·9-s − 2·14-s + 5·16-s − 4·18-s − 3·28-s − 12·29-s + 6·32-s − 6·36-s − 4·37-s − 16·43-s + 49-s − 12·53-s − 4·56-s − 24·58-s + 2·63-s + 7·64-s + 8·67-s − 8·72-s − 8·74-s + 16·79-s − 5·81-s − 32·86-s + 2·98-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s − 0.942·18-s − 0.566·28-s − 2.22·29-s + 1.06·32-s − 36-s − 0.657·37-s − 2.43·43-s + 1/7·49-s − 1.64·53-s − 0.534·56-s − 3.15·58-s + 0.251·63-s + 7/8·64-s + 0.977·67-s − 0.942·72-s − 0.929·74-s + 1.80·79-s − 5/9·81-s − 3.45·86-s + 0.202·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(857500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(54.6749\)
Root analytic conductor: \(2.71923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{857500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 857500,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.71895664384636679391588711627, −7.60556812848957435561623382149, −6.73571587686584923478967639363, −6.66965130292041403123447657278, −6.15190325448345686265108903037, −5.59857578974307577358363282055, −5.12614454716073682582164065811, −5.04607295657079957887442801881, −4.11500666640753372808858013263, −3.77049860941673777607586608067, −3.28327332646103144207952017478, −2.83590121110111537782644859852, −2.07204529193989626181377807359, −1.54150783577442522625509630661, 0, 1.54150783577442522625509630661, 2.07204529193989626181377807359, 2.83590121110111537782644859852, 3.28327332646103144207952017478, 3.77049860941673777607586608067, 4.11500666640753372808858013263, 5.04607295657079957887442801881, 5.12614454716073682582164065811, 5.59857578974307577358363282055, 6.15190325448345686265108903037, 6.66965130292041403123447657278, 6.73571587686584923478967639363, 7.60556812848957435561623382149, 7.71895664384636679391588711627

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