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Properties

Label 4-5696-1.1-c1e2-0-0
Degree $4$
Conductor $5696$
Sign $1$
Analytic cond. $0.363181$
Root an. cond. $0.776302$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 7-s − 2·14-s − 4·16-s + 6·17-s + 3·23-s + 25-s + 2·28-s − 6·31-s + 8·32-s − 12·34-s + 9·41-s − 6·46-s + 47-s − 7·49-s − 2·50-s + 12·62-s − 8·64-s + 12·68-s − 21·71-s − 2·73-s − 10·79-s − 9·81-s − 18·82-s + 14·89-s + 6·92-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.377·7-s − 0.534·14-s − 16-s + 1.45·17-s + 0.625·23-s + 1/5·25-s + 0.377·28-s − 1.07·31-s + 1.41·32-s − 2.05·34-s + 1.40·41-s − 0.884·46-s + 0.145·47-s − 49-s − 0.282·50-s + 1.52·62-s − 64-s + 1.45·68-s − 2.49·71-s − 0.234·73-s − 1.12·79-s − 81-s − 1.98·82-s + 1.48·89-s + 0.625·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5696\)    =    \(2^{6} \cdot 89\)
Sign: $1$
Analytic conductor: \(0.363181\)
Root analytic conductor: \(0.776302\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4747036082\)
\(L(\frac12)\) \(\approx\) \(0.4747036082\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 15 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.ab_i
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
13$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.13.a_au
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.ad_ai
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.29.a_i
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.g_bu
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.37.a_au
41$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.aj_ds
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.ab_bm
53$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.a_ap
59$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.59.a_i
61$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.61.a_da
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.71.v_jm
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.c_es
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.79.k_hb
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.e_du
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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

−11.82992595425393635703302792539, −11.39414924390388464099397038373, −10.77386017949551753345392494641, −10.29332657799234023607454467844, −9.738151895439556723551023257693, −9.097477028253520902546750607443, −8.702076416325054910476466713927, −7.81137035023651695124647336226, −7.59497928114696076685243757952, −6.85758523931776310993182430947, −5.90830598878498044032218021491, −5.13191493880804532693148933802, −4.15389212716158206258524853882, −2.90599355202376937278882597636, −1.44514919932543842686575227267, 1.44514919932543842686575227267, 2.90599355202376937278882597636, 4.15389212716158206258524853882, 5.13191493880804532693148933802, 5.90830598878498044032218021491, 6.85758523931776310993182430947, 7.59497928114696076685243757952, 7.81137035023651695124647336226, 8.702076416325054910476466713927, 9.097477028253520902546750607443, 9.738151895439556723551023257693, 10.29332657799234023607454467844, 10.77386017949551753345392494641, 11.39414924390388464099397038373, 11.82992595425393635703302792539

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