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Properties

Label 4-340e2-1.1-c1e2-0-14
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·5-s + 8·9-s − 4·11-s + 16·15-s + 2·17-s − 8·23-s + 11·25-s − 12·27-s − 10·29-s + 8·31-s + 16·33-s − 10·37-s − 10·41-s − 8·43-s − 32·45-s − 8·51-s − 12·53-s + 16·55-s + 6·61-s + 32·69-s − 16·71-s + 2·73-s − 44·75-s − 24·79-s + 23·81-s − 8·83-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s + 8/3·9-s − 1.20·11-s + 4.13·15-s + 0.485·17-s − 1.66·23-s + 11/5·25-s − 2.30·27-s − 1.85·29-s + 1.43·31-s + 2.78·33-s − 1.64·37-s − 1.56·41-s − 1.21·43-s − 4.77·45-s − 1.12·51-s − 1.64·53-s + 2.15·55-s + 0.768·61-s + 3.85·69-s − 1.89·71-s + 0.234·73-s − 5.08·75-s − 2.70·79-s + 23/9·81-s − 0.878·83-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: \(2\)
Selberg data: \((4,\ 115600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bg
29$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_by
31$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bg
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_by
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_s
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_ey
73$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_c
79$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.79.y_lc
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ak_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

−11.29957606844196519832084896360, −11.27877107106160947426840637420, −10.30870786522520614474114308341, −10.23963843090607582441883671405, −9.930100107253317759111131801498, −8.745592747279040947137132402334, −8.414237457210297435637075050986, −7.84685051837060851989525114925, −7.34155188480238317660416460722, −7.10101440436162860210972239989, −6.22461688446445329395810245473, −5.98814480210351023139675906912, −5.17548995543424279969540039162, −5.06220456657421976569839831891, −4.34851692528485469154626832774, −3.73238614945087440962825918891, −3.07899505296726618871721097361, −1.64123951760712376090462431056, 0, 0, 1.64123951760712376090462431056, 3.07899505296726618871721097361, 3.73238614945087440962825918891, 4.34851692528485469154626832774, 5.06220456657421976569839831891, 5.17548995543424279969540039162, 5.98814480210351023139675906912, 6.22461688446445329395810245473, 7.10101440436162860210972239989, 7.34155188480238317660416460722, 7.84685051837060851989525114925, 8.414237457210297435637075050986, 8.745592747279040947137132402334, 9.930100107253317759111131801498, 10.23963843090607582441883671405, 10.30870786522520614474114308341, 11.27877107106160947426840637420, 11.29957606844196519832084896360

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