Properties

Label 4-507e2-1.1-c1e2-0-1
Degree $4$
Conductor $257049$
Sign $1$
Analytic cond. $16.3896$
Root an. cond. $2.01206$
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 − 3·16-s + 12·17-s − 10·25-s − 4·27-s + 12·29-s − 3·36-s + 8·43-s + 6·48-s − 2·49-s − 24·51-s + 12·53-s − 4·61-s + 7·64-s − 12·68-s + 20·75-s − 16·79-s + 5·81-s − 24·87-s + 10·100-s − 12·101-s − 16·103-s + 24·107-s + 4·108-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s − 3/4·16-s + 2.91·17-s − 2·25-s − 0.769·27-s + 2.22·29-s − 1/2·36-s + 1.21·43-s + 0.866·48-s − 2/7·49-s − 3.36·51-s + 1.64·53-s − 0.512·61-s + 7/8·64-s − 1.45·68-s + 2.30·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s + 100-s − 1.19·101-s − 1.57·103-s + 2.32·107-s + 0.384·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(257049\)    =    \(3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(16.3896\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 257049,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9919123814\)
\(L(\frac12)\) \(\approx\) \(0.9919123814\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−11.36115687945316413102513683604, −10.41858771464550174849576443217, −10.23186052604118683807518466314, −10.10060916897666945627671856311, −9.352195754662219971960634990919, −9.173280372302837284479603894925, −8.212734827287561826163613715312, −8.069517973257505027287831482119, −7.50263456490497252643625573526, −7.01876676244882275910853449372, −6.45326743738505417550753983766, −5.88092650763199403169733671603, −5.52338802841200730223846928180, −5.21583610170390546783438457701, −4.35300108571061840677486504793, −4.14749798025785106783993858289, −3.34537566354701850612787093594, −2.61581680742653857015153061824, −1.50232947470805817130888796036, −0.70031543027443632680126284871, 0.70031543027443632680126284871, 1.50232947470805817130888796036, 2.61581680742653857015153061824, 3.34537566354701850612787093594, 4.14749798025785106783993858289, 4.35300108571061840677486504793, 5.21583610170390546783438457701, 5.52338802841200730223846928180, 5.88092650763199403169733671603, 6.45326743738505417550753983766, 7.01876676244882275910853449372, 7.50263456490497252643625573526, 8.069517973257505027287831482119, 8.212734827287561826163613715312, 9.173280372302837284479603894925, 9.352195754662219971960634990919, 10.10060916897666945627671856311, 10.23186052604118683807518466314, 10.41858771464550174849576443217, 11.36115687945316413102513683604

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