Properties

Label 4-9702e2-1.1-c1e2-0-24
Degree $4$
Conductor $94128804$
Sign $1$
Analytic cond. $6001.73$
Root an. cond. $8.80175$
Motivic weight $1$
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 + 3·4-s − 4·5-s − 4·8-s + 8·10-s + 2·11-s + 8·13-s + 5·16-s − 4·17-s + 4·19-s − 12·20-s − 4·22-s + 4·23-s + 2·25-s − 16·26-s − 8·29-s + 4·31-s − 6·32-s + 8·34-s − 4·37-s − 8·38-s + 16·40-s − 12·41-s − 4·43-s + 6·44-s − 8·46-s − 4·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2.52·10-s + 0.603·11-s + 2.21·13-s + 5/4·16-s − 0.970·17-s + 0.917·19-s − 2.68·20-s − 0.852·22-s + 0.834·23-s + 2/5·25-s − 3.13·26-s − 1.48·29-s + 0.718·31-s − 1.06·32-s + 1.37·34-s − 0.657·37-s − 1.29·38-s + 2.52·40-s − 1.87·41-s − 0.609·43-s + 0.904·44-s − 1.17·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(94128804\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(6001.73\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9702} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 94128804,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−7.46097777734601280385088201311, −7.34631135086647411826343601970, −6.84398101536375019088910068645, −6.82281213986347351775868191746, −6.16149469322688461261825954011, −6.14100495040915870233285696474, −5.41473216242024660065866511989, −5.32633217586220673048508144680, −4.50578519622270047206322049979, −4.35313128952855843786545992903, −3.77095578713166586613405112466, −3.60037246306004207729214130025, −3.17455506569084298960814445469, −3.09015424304817645981087603647, −1.96894898700370322778528809264, −1.96617408269351178127910299745, −1.09552030927412792905760582792, −1.08946590599935230753533481686, 0, 0, 1.08946590599935230753533481686, 1.09552030927412792905760582792, 1.96617408269351178127910299745, 1.96894898700370322778528809264, 3.09015424304817645981087603647, 3.17455506569084298960814445469, 3.60037246306004207729214130025, 3.77095578713166586613405112466, 4.35313128952855843786545992903, 4.50578519622270047206322049979, 5.32633217586220673048508144680, 5.41473216242024660065866511989, 6.14100495040915870233285696474, 6.16149469322688461261825954011, 6.82281213986347351775868191746, 6.84398101536375019088910068645, 7.34631135086647411826343601970, 7.46097777734601280385088201311

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