Properties

Label 4-9702e2-1.1-c1e2-0-22
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 − 4·8-s + 2·11-s + 5·16-s − 4·22-s + 8·23-s − 8·25-s − 6·32-s − 12·37-s − 8·43-s + 6·44-s − 16·46-s + 16·50-s − 12·53-s + 7·64-s − 20·67-s + 24·74-s − 12·79-s + 16·86-s − 8·88-s + 24·92-s − 24·100-s + 24·106-s + 8·107-s − 16·109-s + 36·113-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s + 5/4·16-s − 0.852·22-s + 1.66·23-s − 8/5·25-s − 1.06·32-s − 1.97·37-s − 1.21·43-s + 0.904·44-s − 2.35·46-s + 2.26·50-s − 1.64·53-s + 7/8·64-s − 2.44·67-s + 2.78·74-s − 1.35·79-s + 1.72·86-s − 0.852·88-s + 2.50·92-s − 2.39·100-s + 2.33·106-s + 0.773·107-s − 1.53·109-s + 3.38·113-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$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 162 T^{2} + p^{2} T^{4} \)
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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

−7.55187761301673471673453015967, −7.26721254272450337064002970402, −6.89716801426882001128186732030, −6.70420178923516279929851793442, −6.14287740165238173076002250287, −6.06073444193383960591800673440, −5.56407894486452325254738461243, −5.18085699693998099544086586427, −4.69567332972097635517946725491, −4.52289302762779595756655046820, −3.70150398585820719141554591090, −3.63153274572928700642277542764, −2.98631160963046764950037450601, −2.93805677255495409999842716933, −2.10180590607018621017837666247, −1.84075009332519664123753407761, −1.38208657977308203308057694588, −1.04182615470983694881215387900, 0, 0, 1.04182615470983694881215387900, 1.38208657977308203308057694588, 1.84075009332519664123753407761, 2.10180590607018621017837666247, 2.93805677255495409999842716933, 2.98631160963046764950037450601, 3.63153274572928700642277542764, 3.70150398585820719141554591090, 4.52289302762779595756655046820, 4.69567332972097635517946725491, 5.18085699693998099544086586427, 5.56407894486452325254738461243, 6.06073444193383960591800673440, 6.14287740165238173076002250287, 6.70420178923516279929851793442, 6.89716801426882001128186732030, 7.26721254272450337064002970402, 7.55187761301673471673453015967

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