Properties

Label 4-353e2-1.1-c1e2-0-0
Degree $4$
Conductor $124609$
Sign $1$
Analytic cond. $7.94517$
Root an. cond. $1.67890$
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 − 8·8-s − 4·10-s + 8·11-s − 10·13-s + 8·14-s − 7·16-s + 12·17-s + 2·20-s + 16·22-s + 2·25-s − 20·26-s − 4·28-s − 16·29-s + 4·31-s + 14·32-s + 24·34-s − 8·35-s + 2·37-s + 16·40-s − 8·44-s + 8·49-s + 4·50-s + 10·52-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s − 2.82·8-s − 1.26·10-s + 2.41·11-s − 2.77·13-s + 2.13·14-s − 7/4·16-s + 2.91·17-s + 0.447·20-s + 3.41·22-s + 2/5·25-s − 3.92·26-s − 0.755·28-s − 2.97·29-s + 0.718·31-s + 2.47·32-s + 4.11·34-s − 1.35·35-s + 0.328·37-s + 2.52·40-s − 1.20·44-s + 8/7·49-s + 0.565·50-s + 1.38·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(124609\)    =    \(353^{2}\)
Sign: $1$
Analytic conductor: \(7.94517\)
Root analytic conductor: \(1.67890\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 124609,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.182040401\)
\(L(\frac12)\) \(\approx\) \(2.182040401\)
\(L(\frac{3}{2})\) 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)$
bad353$C_2$ \( 1 - 34 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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

−11.73499242805631641575445202559, −11.68092496626423461898166746544, −11.23251731114294798755689514895, −10.19157766177325350481939323980, −9.571831901308298449791252338454, −9.541692197484461231582911460103, −9.097358121288017269240993430167, −8.303815142601038118375789798015, −7.78875607950524281601984984360, −7.64769654126332614403227401596, −6.92266356100809651694190299694, −6.18257284456624814011486719394, −5.42717165426188793701252918394, −5.17526999242068343149488089779, −4.80194708452025598529834105556, −4.14061019497153244659627801512, −3.67377711730496744681572492732, −3.41519320250482820576957322095, −2.19327171792177506499199293725, −0.896721915054598669896885141755, 0.896721915054598669896885141755, 2.19327171792177506499199293725, 3.41519320250482820576957322095, 3.67377711730496744681572492732, 4.14061019497153244659627801512, 4.80194708452025598529834105556, 5.17526999242068343149488089779, 5.42717165426188793701252918394, 6.18257284456624814011486719394, 6.92266356100809651694190299694, 7.64769654126332614403227401596, 7.78875607950524281601984984360, 8.303815142601038118375789798015, 9.097358121288017269240993430167, 9.541692197484461231582911460103, 9.571831901308298449791252338454, 10.19157766177325350481939323980, 11.23251731114294798755689514895, 11.68092496626423461898166746544, 11.73499242805631641575445202559

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