Properties

Label 4-3822e2-1.1-c1e2-0-5
Degree $4$
Conductor $14607684$
Sign $1$
Analytic cond. $931.398$
Root an. cond. $5.52438$
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s + 4·17-s + 8·23-s + 6·25-s − 4·27-s − 20·29-s − 3·36-s − 12·39-s + 8·43-s − 2·48-s − 8·51-s − 6·52-s − 12·53-s − 4·61-s − 64-s − 4·68-s − 16·69-s − 12·75-s + 5·81-s + 40·87-s − 8·92-s − 6·100-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.970·17-s + 1.66·23-s + 6/5·25-s − 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s + 1.21·43-s − 0.288·48-s − 1.12·51-s − 0.832·52-s − 1.64·53-s − 0.512·61-s − 1/8·64-s − 0.485·68-s − 1.92·69-s − 1.38·75-s + 5/9·81-s + 4.28·87-s − 0.834·92-s − 3/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14607684\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(931.398\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3822} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14607684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.983458005\)
\(L(\frac12)\) \(\approx\) \(1.983458005\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−8.592443095006908719630686474791, −8.508678442950309649005647619017, −7.83144886322286838288380768687, −7.43831754561200452866466859406, −7.18606805449285450629387747142, −6.99727455278791735341603897689, −6.11725242134247229214825955898, −6.06845525461585603333445897033, −5.81251956109654462293586506546, −5.34711438141615528898344326510, −4.97300766978274485378060510492, −4.57151468145937373871369189002, −4.21043966905202273620386678418, −3.48681924873775414111823132062, −3.43753072507114542699678702954, −3.01230306631566244689649768439, −1.79593973989094017552982170993, −1.79502897079381486437847821399, −0.844615223775748613531646780089, −0.63532087720513272905995758472, 0.63532087720513272905995758472, 0.844615223775748613531646780089, 1.79502897079381486437847821399, 1.79593973989094017552982170993, 3.01230306631566244689649768439, 3.43753072507114542699678702954, 3.48681924873775414111823132062, 4.21043966905202273620386678418, 4.57151468145937373871369189002, 4.97300766978274485378060510492, 5.34711438141615528898344326510, 5.81251956109654462293586506546, 6.06845525461585603333445897033, 6.11725242134247229214825955898, 6.99727455278791735341603897689, 7.18606805449285450629387747142, 7.43831754561200452866466859406, 7.83144886322286838288380768687, 8.508678442950309649005647619017, 8.592443095006908719630686474791

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