Properties

Label 4-495e2-1.1-c1e2-0-5
Degree $4$
Conductor $245025$
Sign $1$
Analytic cond. $15.6230$
Root an. cond. $1.98811$
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  + 2·2-s + 4-s + 2·5-s − 4·7-s + 4·10-s + 2·11-s − 8·14-s + 16-s + 8·17-s − 8·19-s + 2·20-s + 4·22-s + 8·23-s + 3·25-s − 4·28-s + 4·29-s − 2·32-s + 16·34-s − 8·35-s + 12·37-s − 16·38-s − 4·41-s − 12·43-s + 2·44-s + 16·46-s + 8·47-s + 6·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s + 1.26·10-s + 0.603·11-s − 2.13·14-s + 1/4·16-s + 1.94·17-s − 1.83·19-s + 0.447·20-s + 0.852·22-s + 1.66·23-s + 3/5·25-s − 0.755·28-s + 0.742·29-s − 0.353·32-s + 2.74·34-s − 1.35·35-s + 1.97·37-s − 2.59·38-s − 0.624·41-s − 1.82·43-s + 0.301·44-s + 2.35·46-s + 1.16·47-s + 6/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(245025\)    =    \(3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(15.6230\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{495} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 245025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.639022901\)
\(L(\frac12)\) \(\approx\) \(3.639022901\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−11.24319790166052915197437532757, −10.51843968414512898332519903296, −10.34063493019675716825039313213, −10.00169364499106846710884884196, −9.296498886467154434148831168775, −9.169724094385808836303108789430, −8.602383164944062514213902740200, −7.969461359899934356764838845175, −7.29379222370783295255333182248, −6.83591980305966662691079565393, −6.20289516130001992071945017691, −6.16659143320813207864795445591, −5.52114127546218421109580391860, −4.97648828837409152129173765477, −4.50391485936276240651511954468, −3.92490376183820598138637798674, −3.18629491234324629698620261642, −3.08210702735664895319434708931, −2.07132368734417524878830890541, −0.992081852202220186506946909778, 0.992081852202220186506946909778, 2.07132368734417524878830890541, 3.08210702735664895319434708931, 3.18629491234324629698620261642, 3.92490376183820598138637798674, 4.50391485936276240651511954468, 4.97648828837409152129173765477, 5.52114127546218421109580391860, 6.16659143320813207864795445591, 6.20289516130001992071945017691, 6.83591980305966662691079565393, 7.29379222370783295255333182248, 7.969461359899934356764838845175, 8.602383164944062514213902740200, 9.169724094385808836303108789430, 9.296498886467154434148831168775, 10.00169364499106846710884884196, 10.34063493019675716825039313213, 10.51843968414512898332519903296, 11.24319790166052915197437532757

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