Properties

Label 4-882e2-1.1-c3e2-0-28
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
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  + 2·2-s + 6·5-s − 8·8-s + 12·10-s + 30·11-s + 4·13-s − 16·16-s + 66·17-s + 52·19-s + 60·22-s + 114·23-s + 125·25-s + 8·26-s − 144·29-s + 196·31-s + 132·34-s + 286·37-s + 104·38-s − 48·40-s + 756·41-s + 328·43-s + 228·46-s − 228·47-s + 250·50-s − 348·53-s + 180·55-s − 288·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.536·5-s − 0.353·8-s + 0.379·10-s + 0.822·11-s + 0.0853·13-s − 1/4·16-s + 0.941·17-s + 0.627·19-s + 0.581·22-s + 1.03·23-s + 25-s + 0.0603·26-s − 0.922·29-s + 1.13·31-s + 0.665·34-s + 1.27·37-s + 0.443·38-s − 0.189·40-s + 2.87·41-s + 1.16·43-s + 0.730·46-s − 0.707·47-s + 0.707·50-s − 0.901·53-s + 0.441·55-s − 0.652·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.408380115\)
\(L(\frac12)\) \(\approx\) \(7.408380115\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 66 T - 557 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 52 T - 4155 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 114 T + 829 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 196 T + 8625 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 286 T + 31143 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 378 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 228 T - 51839 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 348 T - 27773 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 348 T - 84275 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 106 T - 215745 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 630 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 1042 T + 696747 T^{2} - 1042 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 88 T - 485295 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1440 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1374 T + 1182907 T^{2} - 1374 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
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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

−9.726797115736370527362931270699, −9.509378862932006278287501647299, −9.092074987881284227973494814722, −9.089030824223987748543971069676, −7.991330549981919406288469274755, −7.989156998121775375580681715883, −7.35057479006364634303717958904, −6.92435915486380404192361273043, −6.30646798360923233614867252891, −5.95511027388913056515955278179, −5.75134536372381511610656400731, −4.99242889282001835768028142554, −4.66425588011873060933917611903, −4.22643940687768288615656634835, −3.55392066360507094528170575125, −3.08441164076010918014194814156, −2.64899658337377096953648270649, −1.89033874185499424632568491923, −0.941005237095029963353314272245, −0.845934592315256702655663330449, 0.845934592315256702655663330449, 0.941005237095029963353314272245, 1.89033874185499424632568491923, 2.64899658337377096953648270649, 3.08441164076010918014194814156, 3.55392066360507094528170575125, 4.22643940687768288615656634835, 4.66425588011873060933917611903, 4.99242889282001835768028142554, 5.75134536372381511610656400731, 5.95511027388913056515955278179, 6.30646798360923233614867252891, 6.92435915486380404192361273043, 7.35057479006364634303717958904, 7.989156998121775375580681715883, 7.991330549981919406288469274755, 9.089030824223987748543971069676, 9.092074987881284227973494814722, 9.509378862932006278287501647299, 9.726797115736370527362931270699

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