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Properties

Label 4-340e2-1.1-c1e2-0-16
Degree $4$
Conductor $115600$
Sign $-1$
Analytic cond. $7.37075$
Root an. cond. $1.64769$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s + 4·13-s − 16-s − 6·17-s − 8·19-s − 25-s + 4·26-s + 5·32-s − 6·34-s − 8·38-s − 22·43-s + 2·47-s + 8·49-s − 50-s − 4·52-s − 8·53-s − 8·59-s + 7·64-s + 14·67-s + 6·68-s + 8·76-s − 9·81-s − 2·83-s − 22·86-s + 2·94-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1.10·13-s − 1/4·16-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.784·26-s + 0.883·32-s − 1.02·34-s − 1.29·38-s − 3.35·43-s + 0.291·47-s + 8/7·49-s − 0.141·50-s − 0.554·52-s − 1.09·53-s − 1.04·59-s + 7/8·64-s + 1.71·67-s + 0.727·68-s + 0.917·76-s − 81-s − 0.219·83-s − 2.37·86-s + 0.206·94-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: yes
Self-dual: yes
Analytic rank: \(1\)
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$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ae_o
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.i_bm
23$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.23.a_abo
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.31.a_i
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.41.a_cg
43$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.w_hy
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.ac_o
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.i_eo
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.i_cs
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.ao_gs
71$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \) 2.71.a_cu
73$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.73.a_abe
79$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \) 2.79.a_aeq
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.c_ac
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−9.136503748669288671350125199405, −8.670662191513768568756438002112, −8.335924935947455064066194787716, −8.025461497321618315518428203691, −6.84690166468947991685472434175, −6.70555673220239224394984954620, −6.15576634029471464930388930372, −5.64583776038304322414395614853, −4.91633121139425357435822763962, −4.47346975187280301993519899494, −3.94730448018956220296682974365, −3.42017911670120886570600770541, −2.55553166155643940516752807561, −1.69208974574332137498775334130, 0, 1.69208974574332137498775334130, 2.55553166155643940516752807561, 3.42017911670120886570600770541, 3.94730448018956220296682974365, 4.47346975187280301993519899494, 4.91633121139425357435822763962, 5.64583776038304322414395614853, 6.15576634029471464930388930372, 6.70555673220239224394984954620, 6.84690166468947991685472434175, 8.025461497321618315518428203691, 8.335924935947455064066194787716, 8.670662191513768568756438002112, 9.136503748669288671350125199405

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