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Properties

Label 4-340e2-1.1-c1e2-0-12
Degree $4$
Conductor $115600$
Sign $1$
Analytic cond. $7.37075$
Root an. cond. $1.64769$
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  + 2·2-s + 2·4-s + 4·5-s + 8·10-s + 2·13-s − 4·16-s + 8·17-s + 8·20-s + 11·25-s + 4·26-s − 20·29-s − 8·32-s + 16·34-s − 10·37-s + 22·50-s + 4·52-s + 18·53-s − 40·58-s − 8·64-s + 8·65-s + 16·68-s + 10·73-s − 20·74-s − 16·80-s − 9·81-s + 32·85-s − 10·97-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s + 2.52·10-s + 0.554·13-s − 16-s + 1.94·17-s + 1.78·20-s + 11/5·25-s + 0.784·26-s − 3.71·29-s − 1.41·32-s + 2.74·34-s − 1.64·37-s + 3.11·50-s + 0.554·52-s + 2.47·53-s − 5.25·58-s − 64-s + 0.992·65-s + 1.94·68-s + 1.17·73-s − 2.32·74-s − 1.78·80-s − 81-s + 3.47·85-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(115600\)    =    \(2^{4} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(7.37075\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 115600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.460437619\)
\(L(\frac12)\) \(\approx\) \(4.460437619\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_2$ \( 1 - 8 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.ac_c
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.as_gg
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.k_by
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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

−11.98002012353268937319550724097, −11.38961192761939788018784619860, −10.85443755500901586199281316203, −10.51966915853878124657601935344, −9.794057888582112997668458879022, −9.656841411070446704639756335200, −9.014510985185415028167487529491, −8.709251911422961734965856425360, −7.85162343241887698740491356459, −7.23475224260280698281711463759, −6.82549173827902152596767672483, −6.15080366012953773311030885016, −5.52798194217275465272735371821, −5.52770991282800101759056804829, −5.18416447527091488508098656587, −3.99946772094043422010094958752, −3.67376613256604571084800990535, −2.97182785072440518286992929891, −2.14612365211137396266405409380, −1.50483302313949655495933201708, 1.50483302313949655495933201708, 2.14612365211137396266405409380, 2.97182785072440518286992929891, 3.67376613256604571084800990535, 3.99946772094043422010094958752, 5.18416447527091488508098656587, 5.52770991282800101759056804829, 5.52798194217275465272735371821, 6.15080366012953773311030885016, 6.82549173827902152596767672483, 7.23475224260280698281711463759, 7.85162343241887698740491356459, 8.709251911422961734965856425360, 9.014510985185415028167487529491, 9.656841411070446704639756335200, 9.794057888582112997668458879022, 10.51966915853878124657601935344, 10.85443755500901586199281316203, 11.38961192761939788018784619860, 11.98002012353268937319550724097

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