Properties

Label 4-336e2-1.1-c3e2-0-5
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $393.016$
Root an. cond. $4.45248$
Motivic weight $3$
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  + 3·3-s + 3·5-s + 7·7-s − 15·11-s − 128·13-s + 9·15-s − 84·17-s − 16·19-s + 21·21-s − 84·23-s + 125·25-s − 27·27-s − 594·29-s − 253·31-s − 45·33-s + 21·35-s + 316·37-s − 384·39-s + 720·41-s − 52·43-s − 30·47-s − 294·49-s − 252·51-s − 363·53-s − 45·55-s − 48·57-s − 15·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.268·5-s + 0.377·7-s − 0.411·11-s − 2.73·13-s + 0.154·15-s − 1.19·17-s − 0.193·19-s + 0.218·21-s − 0.761·23-s + 25-s − 0.192·27-s − 3.80·29-s − 1.46·31-s − 0.237·33-s + 0.101·35-s + 1.40·37-s − 1.57·39-s + 2.74·41-s − 0.184·43-s − 0.0931·47-s − 6/7·49-s − 0.691·51-s − 0.940·53-s − 0.110·55-s − 0.111·57-s − 0.0330·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(393.016\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{336} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7734167502\)
\(L(\frac12)\) \(\approx\) \(0.7734167502\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p T + p^{2} T^{2} \)
7$C_2$ \( 1 - p T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 64 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 16 T - 6603 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 297 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 253 T + 34218 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 360 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 26 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 363 T - 17108 T^{2} + 363 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 370 T - 163863 T^{2} + 370 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 467 T - 274950 T^{2} - 467 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 477 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 906 T + 115867 T^{2} + 906 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )^{2} \)
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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

−11.56151736026734699130438821332, −10.88440997226486244201935453340, −10.52245558081056606533906672721, −9.615631854361670187086026690678, −9.483828788112453373329368424623, −9.297560201848005976603054839226, −8.622555572395360024345210052951, −7.74024914171949461009663672447, −7.68029492314453581041215346125, −7.28160484462312649224124833904, −6.68118625159355183565558342623, −5.65557555992782630641101166463, −5.65373929054438857931024604123, −4.67858363659003933092439063930, −4.46211532461801271616658170915, −3.64095791833302153247079519182, −2.76952250924329352133802909574, −2.18006190955461941349133614785, −1.91406231343702784776374349737, −0.27613058830685716171005661764, 0.27613058830685716171005661764, 1.91406231343702784776374349737, 2.18006190955461941349133614785, 2.76952250924329352133802909574, 3.64095791833302153247079519182, 4.46211532461801271616658170915, 4.67858363659003933092439063930, 5.65373929054438857931024604123, 5.65557555992782630641101166463, 6.68118625159355183565558342623, 7.28160484462312649224124833904, 7.68029492314453581041215346125, 7.74024914171949461009663672447, 8.622555572395360024345210052951, 9.297560201848005976603054839226, 9.483828788112453373329368424623, 9.615631854361670187086026690678, 10.52245558081056606533906672721, 10.88440997226486244201935453340, 11.56151736026734699130438821332

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