Properties

Label 4-980e2-1.1-c1e2-0-7
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 4·5-s − 8·10-s + 2·13-s − 4·16-s − 6·17-s + 8·20-s + 11·25-s − 4·26-s + 8·32-s + 12·34-s − 14·37-s + 16·41-s − 22·50-s + 4·52-s + 18·53-s − 24·61-s − 8·64-s + 8·65-s − 12·68-s + 22·73-s + 28·74-s − 16·80-s − 9·81-s − 32·82-s − 24·85-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 2.52·10-s + 0.554·13-s − 16-s − 1.45·17-s + 1.78·20-s + 11/5·25-s − 0.784·26-s + 1.41·32-s + 2.05·34-s − 2.30·37-s + 2.49·41-s − 3.11·50-s + 0.554·52-s + 2.47·53-s − 3.07·61-s − 64-s + 0.992·65-s − 1.45·68-s + 2.57·73-s + 3.25·74-s − 1.78·80-s − 81-s − 3.53·82-s − 2.60·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.313632531\)
\(L(\frac12)\) \(\approx\) \(1.313632531\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.29224204712433226441981172000, −9.542060030139620300768090574470, −9.365110741029910619256331736536, −9.121255509298382547657694989555, −8.630111632380304207579939955611, −8.437217943845425309188871866901, −7.78808704832649488271447688475, −7.23816351189898264554266500128, −6.89026133850784528458626812718, −6.45628598320633701864739215954, −6.14316731950638965947810085938, −5.38113094412093304613718265126, −5.34156922089182953656772482947, −4.34389401237154677086073670140, −4.17706873814389346989570308742, −3.08469338907629900369255435504, −2.53773224515776501301886622428, −1.95477878322030653348419473792, −1.58319308723117379137868562610, −0.68294368054670175838951307942, 0.68294368054670175838951307942, 1.58319308723117379137868562610, 1.95477878322030653348419473792, 2.53773224515776501301886622428, 3.08469338907629900369255435504, 4.17706873814389346989570308742, 4.34389401237154677086073670140, 5.34156922089182953656772482947, 5.38113094412093304613718265126, 6.14316731950638965947810085938, 6.45628598320633701864739215954, 6.89026133850784528458626812718, 7.23816351189898264554266500128, 7.78808704832649488271447688475, 8.437217943845425309188871866901, 8.630111632380304207579939955611, 9.121255509298382547657694989555, 9.365110741029910619256331736536, 9.542060030139620300768090574470, 10.29224204712433226441981172000

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