Properties

Label 4-490e2-1.1-c1e2-0-1
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $15.3089$
Root an. cond. $1.97804$
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-s + 5-s + 8-s + 3·9-s − 10-s − 4·11-s − 12·13-s − 16-s − 2·17-s − 3·18-s + 4·22-s + 12·26-s + 12·29-s − 8·31-s + 2·34-s + 10·37-s + 40-s + 4·41-s + 8·43-s + 3·45-s − 8·47-s + 2·53-s − 4·55-s − 12·58-s + 8·59-s + 14·61-s + 8·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s + 0.353·8-s + 9-s − 0.316·10-s − 1.20·11-s − 3.32·13-s − 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.852·22-s + 2.35·26-s + 2.22·29-s − 1.43·31-s + 0.342·34-s + 1.64·37-s + 0.158·40-s + 0.624·41-s + 1.21·43-s + 0.447·45-s − 1.16·47-s + 0.274·53-s − 0.539·55-s − 1.57·58-s + 1.04·59-s + 1.79·61-s + 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15.3089\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 240100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7960676288\)
\(L(\frac12)\) \(\approx\) \(0.7960676288\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.91449885548421353299368692065, −10.58964009007528861941458198028, −10.07705188758104215579701612426, −9.830163187080914398606857518466, −9.624139854098159614713287465549, −9.144393881940340659438393943784, −8.451913079789703340567323713260, −7.993270654708154320940015719060, −7.50779843881752911088938721503, −7.23653273850540955255724464992, −6.86651061664799124394628266167, −6.17434434099249059185973251377, −5.27449087921738445375888038421, −5.16357616770303555663819537530, −4.46419082400885660387656749542, −4.20461696965546731226072918316, −2.79429567334652041924205310628, −2.56922720508448546235304923088, −1.91962626774435033059449131011, −0.59109521942451079983750827339, 0.59109521942451079983750827339, 1.91962626774435033059449131011, 2.56922720508448546235304923088, 2.79429567334652041924205310628, 4.20461696965546731226072918316, 4.46419082400885660387656749542, 5.16357616770303555663819537530, 5.27449087921738445375888038421, 6.17434434099249059185973251377, 6.86651061664799124394628266167, 7.23653273850540955255724464992, 7.50779843881752911088938721503, 7.993270654708154320940015719060, 8.451913079789703340567323713260, 9.144393881940340659438393943784, 9.624139854098159614713287465549, 9.830163187080914398606857518466, 10.07705188758104215579701612426, 10.58964009007528861941458198028, 10.91449885548421353299368692065

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