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Properties

Label 4-281216-1.1-c1e2-0-4
Degree $4$
Conductor $281216$
Sign $-1$
Analytic cond. $17.9305$
Root an. cond. $2.05777$
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  + 2·5-s − 9-s − 13-s − 6·17-s − 3·25-s + 8·29-s − 6·37-s − 16·41-s − 2·45-s − 5·49-s − 4·53-s − 16·61-s − 2·65-s + 4·73-s − 8·81-s − 12·85-s − 20·89-s − 8·97-s + 8·101-s + 26·109-s − 20·113-s + 117-s − 10·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s − 0.277·13-s − 1.45·17-s − 3/5·25-s + 1.48·29-s − 0.986·37-s − 2.49·41-s − 0.298·45-s − 5/7·49-s − 0.549·53-s − 2.04·61-s − 0.248·65-s + 0.468·73-s − 8/9·81-s − 1.30·85-s − 2.11·89-s − 0.812·97-s + 0.796·101-s + 2.49·109-s − 1.88·113-s + 0.0924·117-s − 0.909·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(281216\)    =    \(2^{7} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(17.9305\)
Root analytic conductor: \(2.05777\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 281216,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
13$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.ac_h
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.g_bb
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ai_cs
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.g_bv
41$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.q_fa
43$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.43.a_ax
47$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \) 2.47.a_acp
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.59.a_bi
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.67.a_adi
71$C_2^2$ \( 1 - 123 T^{2} + p^{2} T^{4} \) 2.71.a_aet
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.83.a_fe
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.u_kc
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.i_hy
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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

−8.660767169296377088430904953954, −8.258283246257513980875743405201, −7.83196105590512252969556669260, −6.95315098392942841120221022780, −6.78818224990205704781092057818, −6.31483052178314498203629314991, −5.73278526352893523877509085751, −5.29106642713298523747034230740, −4.63963009312516773161808090632, −4.34216952994272400925609127105, −3.34278660076602225747753433367, −2.91295046418483861454133703113, −2.06300003098660776435788514770, −1.60093497804832542144117980651, 0, 1.60093497804832542144117980651, 2.06300003098660776435788514770, 2.91295046418483861454133703113, 3.34278660076602225747753433367, 4.34216952994272400925609127105, 4.63963009312516773161808090632, 5.29106642713298523747034230740, 5.73278526352893523877509085751, 6.31483052178314498203629314991, 6.78818224990205704781092057818, 6.95315098392942841120221022780, 7.83196105590512252969556669260, 8.258283246257513980875743405201, 8.660767169296377088430904953954

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