Properties

Label 4-2300e2-1.1-c0e2-0-0
Degree $4$
Conductor $5290000$
Sign $1$
Analytic cond. $1.31755$
Root an. cond. $1.07137$
Motivic weight $0$
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  − 2·9-s + 2·29-s − 2·31-s − 2·41-s + 49-s + 2·59-s − 2·71-s + 3·81-s − 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s + 2·29-s − 2·31-s − 2·41-s + 49-s + 2·59-s − 2·71-s + 3·81-s − 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5290000\)    =    \(2^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1.31755\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5290000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8336366468\)
\(L(\frac12)\) \(\approx\) \(0.8336366468\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2^2$ \( 1 - T^{2} + T^{4} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.303587537063022424143505501091, −8.899046954412011277903195114486, −8.471149978450218216582872694376, −8.332372418328144980826522453140, −8.180606103796774650434763485218, −7.32724748424484009276862803556, −7.01102658085386639906683169096, −6.85654343407979206248796038477, −6.08720100208354433207255720613, −5.82257757738244732433199331914, −5.60061225610921575611939086583, −5.04560658088589069696271469783, −4.72698983840566093282233521706, −4.12455704756833523661685995081, −3.50861133757042811047355588403, −3.22088148634155105821518002431, −2.74578055731533913790012377181, −2.23459891576534686621150515256, −1.64777480925348351668197014217, −0.61456437740165562383585686219, 0.61456437740165562383585686219, 1.64777480925348351668197014217, 2.23459891576534686621150515256, 2.74578055731533913790012377181, 3.22088148634155105821518002431, 3.50861133757042811047355588403, 4.12455704756833523661685995081, 4.72698983840566093282233521706, 5.04560658088589069696271469783, 5.60061225610921575611939086583, 5.82257757738244732433199331914, 6.08720100208354433207255720613, 6.85654343407979206248796038477, 7.01102658085386639906683169096, 7.32724748424484009276862803556, 8.180606103796774650434763485218, 8.332372418328144980826522453140, 8.471149978450218216582872694376, 8.899046954412011277903195114486, 9.303587537063022424143505501091

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