Properties

Label 4-30752-1.1-c1e2-0-4
Degree $4$
Conductor $30752$
Sign $-1$
Analytic cond. $1.96077$
Root an. cond. $1.18333$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·5-s + 8-s − 6·9-s − 4·10-s + 4·13-s + 16-s − 12·17-s − 6·18-s − 4·20-s + 2·25-s + 4·26-s + 4·29-s + 32-s − 12·34-s − 6·36-s + 20·37-s − 4·40-s − 12·41-s + 24·45-s − 14·49-s + 2·50-s + 4·52-s − 12·53-s + 4·58-s − 12·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 2·9-s − 1.26·10-s + 1.10·13-s + 1/4·16-s − 2.91·17-s − 1.41·18-s − 0.894·20-s + 2/5·25-s + 0.784·26-s + 0.742·29-s + 0.176·32-s − 2.05·34-s − 36-s + 3.28·37-s − 0.632·40-s − 1.87·41-s + 3.57·45-s − 2·49-s + 0.282·50-s + 0.554·52-s − 1.64·53-s + 0.525·58-s − 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30752\)    =    \(2^{5} \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(1.96077\)
Root analytic conductor: \(1.18333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 30752,\ (\ :1/2, 1/2),\ -1)\)

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

−10.72587641522723560036260231629, −9.557473081917962507344751915689, −9.056498267458590117851162621111, −8.371776216834959175000943223311, −8.156371355658038539653222782109, −7.76857292337562945056886541715, −6.72038595262299226278216985203, −6.32858579420662158347624019611, −5.94945525164915580359156684875, −4.74203912193695697715499891246, −4.55787579523055807179904019114, −3.68623363482399645957944200957, −3.16188554299090604324575557253, −2.29075936305574985962112760644, 0, 2.29075936305574985962112760644, 3.16188554299090604324575557253, 3.68623363482399645957944200957, 4.55787579523055807179904019114, 4.74203912193695697715499891246, 5.94945525164915580359156684875, 6.32858579420662158347624019611, 6.72038595262299226278216985203, 7.76857292337562945056886541715, 8.156371355658038539653222782109, 8.371776216834959175000943223311, 9.056498267458590117851162621111, 9.557473081917962507344751915689, 10.72587641522723560036260231629

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