Properties

Label 4-1176e2-1.1-c3e2-0-5
Degree $4$
Conductor $1382976$
Sign $1$
Analytic cond. $4814.45$
Root an. cond. $8.32984$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 14·5-s + 27·9-s + 18·11-s − 48·13-s + 84·15-s − 34·17-s + 16·19-s + 110·23-s + 74·25-s − 108·27-s + 212·29-s + 136·31-s − 108·33-s − 24·37-s + 288·39-s − 694·41-s − 584·43-s − 378·45-s + 316·47-s + 204·51-s + 560·53-s − 252·55-s − 96·57-s + 492·59-s + 604·61-s + 672·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.25·5-s + 9-s + 0.493·11-s − 1.02·13-s + 1.44·15-s − 0.485·17-s + 0.193·19-s + 0.997·23-s + 0.591·25-s − 0.769·27-s + 1.35·29-s + 0.787·31-s − 0.569·33-s − 0.106·37-s + 1.18·39-s − 2.64·41-s − 2.07·43-s − 1.25·45-s + 0.980·47-s + 0.560·51-s + 1.45·53-s − 0.617·55-s − 0.223·57-s + 1.08·59-s + 1.26·61-s + 1.28·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4814.45\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1176} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1382976,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 18 T + 1150 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4262 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 2 p T - 4222 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 16 T + 10950 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 110 T + 18686 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 212 T + 57182 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 136 T + 61374 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 694 T + 243914 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 584 T + 232950 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 316 T + 231902 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 560 T + 369782 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 492 T + 351622 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 604 T + 406398 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1020 T + 843926 T^{2} + 1020 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1710 T + 1336222 T^{2} + 1710 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1312 T + 1201998 T^{2} - 1312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 556 T + 751134 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 264 T + 979750 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 70 T - 186262 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 136 T + 1812270 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.048460658595669905392439427488, −8.812646538320868374070556456041, −8.339222708360629533670222984208, −7.965220564101531614735014827088, −7.39832221676471024988725581102, −6.99295489768284905312888534062, −6.75858535283397507011805079103, −6.47088143551069849894881309445, −5.77136928920602632363721288492, −5.12904767227584886194638138037, −4.96672795103551478098582388337, −4.57670215343281485437865560823, −3.94428898223484441599439630112, −3.62589118794414131788598902065, −2.90589306189702290952525937949, −2.38753439939022871867659373608, −1.41683978304470014556392986697, −0.983120344766232037368998830666, 0, 0, 0.983120344766232037368998830666, 1.41683978304470014556392986697, 2.38753439939022871867659373608, 2.90589306189702290952525937949, 3.62589118794414131788598902065, 3.94428898223484441599439630112, 4.57670215343281485437865560823, 4.96672795103551478098582388337, 5.12904767227584886194638138037, 5.77136928920602632363721288492, 6.47088143551069849894881309445, 6.75858535283397507011805079103, 6.99295489768284905312888534062, 7.39832221676471024988725581102, 7.965220564101531614735014827088, 8.339222708360629533670222984208, 8.812646538320868374070556456041, 9.048460658595669905392439427488

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