Properties

Label 4-980e2-1.1-c1e2-0-25
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 $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s + 3·9-s − 4·10-s − 6·13-s − 4·16-s + 6·17-s − 6·18-s + 4·20-s + 3·25-s + 12·26-s − 2·29-s + 8·32-s − 12·34-s + 6·36-s − 12·41-s + 6·45-s − 6·50-s − 12·52-s − 20·53-s + 4·58-s − 8·64-s − 12·65-s + 12·68-s − 12·73-s − 8·80-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.894·5-s + 9-s − 1.26·10-s − 1.66·13-s − 16-s + 1.45·17-s − 1.41·18-s + 0.894·20-s + 3/5·25-s + 2.35·26-s − 0.371·29-s + 1.41·32-s − 2.05·34-s + 36-s − 1.87·41-s + 0.894·45-s − 0.848·50-s − 1.66·52-s − 2.74·53-s + 0.525·58-s − 64-s − 1.48·65-s + 1.45·68-s − 1.40·73-s − 0.894·80-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: $\chi_{960400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
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

−7.80457391816395721714669401072, −7.56848203557801886905576002740, −7.30703478828022800288631659705, −6.66812515626338690414111791742, −6.46027048767020078983890297601, −5.69074574951383439724683067976, −5.27291442121198189570792311479, −4.68059577608072660097150741909, −4.49258853804675338518966926948, −3.46612148832537381816016895570, −3.00378692647977841566927386749, −2.19920226028059021829079345585, −1.70898778880961935771596537039, −1.18708333743445212285166420152, 0, 1.18708333743445212285166420152, 1.70898778880961935771596537039, 2.19920226028059021829079345585, 3.00378692647977841566927386749, 3.46612148832537381816016895570, 4.49258853804675338518966926948, 4.68059577608072660097150741909, 5.27291442121198189570792311479, 5.69074574951383439724683067976, 6.46027048767020078983890297601, 6.66812515626338690414111791742, 7.30703478828022800288631659705, 7.56848203557801886905576002740, 7.80457391816395721714669401072

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