Properties

Label 4-9702e2-1.1-c1e2-0-13
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 $0$

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 + 8·13-s + 5·16-s − 6·17-s − 4·22-s − 3·25-s + 16·26-s − 4·29-s + 8·31-s + 6·32-s − 12·34-s − 8·37-s + 18·41-s + 8·43-s − 6·44-s + 8·47-s − 6·50-s + 24·52-s − 8·53-s − 8·58-s − 8·59-s − 8·61-s + 16·62-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s + 2.21·13-s + 5/4·16-s − 1.45·17-s − 0.852·22-s − 3/5·25-s + 3.13·26-s − 0.742·29-s + 1.43·31-s + 1.06·32-s − 2.05·34-s − 1.31·37-s + 2.81·41-s + 1.21·43-s − 0.904·44-s + 1.16·47-s − 0.848·50-s + 3.32·52-s − 1.09·53-s − 1.05·58-s − 1.04·59-s − 1.02·61-s + 2.03·62-s + 7/8·64-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: \(0\)
Selberg data: \((4,\ 94128804,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.19251046\)
\(L(\frac12)\) \(\approx\) \(11.19251046\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + p T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 218 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 215 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 214 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−7.64132764128881286666852938513, −7.63400187531291111818990007401, −7.04292896380318220424296161330, −6.53219424750734388994783365266, −6.35613724994122282872227313561, −6.22423912797179634007853925904, −5.70504167314519613392573411436, −5.62981980913128775645604042989, −4.95189172853857840528190969235, −4.78225555064384071681650761989, −4.25570878043122212463295123386, −4.12501175998146233995289112047, −3.56120120022999435142877026996, −3.48759344223791640113922230250, −2.89589189490581160559717492020, −2.45854907330576115931757060400, −2.03349503542897661945235423405, −1.81205445911148969313546865183, −0.925283165064081036981958238165, −0.66986247532245083098613590951, 0.66986247532245083098613590951, 0.925283165064081036981958238165, 1.81205445911148969313546865183, 2.03349503542897661945235423405, 2.45854907330576115931757060400, 2.89589189490581160559717492020, 3.48759344223791640113922230250, 3.56120120022999435142877026996, 4.12501175998146233995289112047, 4.25570878043122212463295123386, 4.78225555064384071681650761989, 4.95189172853857840528190969235, 5.62981980913128775645604042989, 5.70504167314519613392573411436, 6.22423912797179634007853925904, 6.35613724994122282872227313561, 6.53219424750734388994783365266, 7.04292896380318220424296161330, 7.63400187531291111818990007401, 7.64132764128881286666852938513

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