Properties

Label 4-8624e2-1.1-c1e2-0-1
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 9-s − 2·11-s − 2·13-s − 8·15-s − 4·17-s + 4·19-s + 4·23-s + 4·25-s − 6·27-s − 6·29-s + 8·31-s − 4·33-s − 16·37-s − 4·39-s − 8·41-s + 4·45-s + 4·47-s − 8·51-s − 4·53-s + 8·55-s + 8·57-s + 14·59-s + 18·61-s + 8·65-s − 14·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 1/3·9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s − 0.970·17-s + 0.917·19-s + 0.834·23-s + 4/5·25-s − 1.15·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 2.63·37-s − 0.640·39-s − 1.24·41-s + 0.596·45-s + 0.583·47-s − 1.12·51-s − 0.549·53-s + 1.07·55-s + 1.05·57-s + 1.82·59-s + 2.30·61-s + 0.992·65-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8624} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7201797175\)
\(L(\frac12)\) \(\approx\) \(0.7201797175\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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

−7.86235093057928049345003915773, −7.74607036958580384323843468894, −7.35909382414697767814775652248, −7.08172930204316957374182286745, −6.67644969272791938895610773565, −6.52995438067487670437274869847, −5.65874149684560381708502473700, −5.50903971801855883138143774186, −4.99858711854964416335235212352, −4.92830390420495265771885251492, −4.21788074052587846924897411712, −3.99661333331763794510126724146, −3.51100906737367182630481271256, −3.40177385325507443005749039330, −2.85495645664312258994843504940, −2.65649552518627498150037535953, −2.06826634758642928235302460192, −1.71007699948514299823456004611, −0.799094839644138814639646614783, −0.22692525678679928252089930112, 0.22692525678679928252089930112, 0.799094839644138814639646614783, 1.71007699948514299823456004611, 2.06826634758642928235302460192, 2.65649552518627498150037535953, 2.85495645664312258994843504940, 3.40177385325507443005749039330, 3.51100906737367182630481271256, 3.99661333331763794510126724146, 4.21788074052587846924897411712, 4.92830390420495265771885251492, 4.99858711854964416335235212352, 5.50903971801855883138143774186, 5.65874149684560381708502473700, 6.52995438067487670437274869847, 6.67644969272791938895610773565, 7.08172930204316957374182286745, 7.35909382414697767814775652248, 7.74607036958580384323843468894, 7.86235093057928049345003915773

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