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Properties

Label 4-91e2-1.1-c1e2-0-4
Degree $4$
Conductor $8281$
Sign $1$
Analytic cond. $0.528003$
Root an. cond. $0.852431$
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  − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 5·7-s − 3·8-s − 3·9-s + 9·10-s − 8·12-s − 2·13-s + 15·14-s + 6·15-s + 3·16-s − 6·17-s + 9·18-s − 12·20-s + 10·21-s + 6·24-s + 25-s + 6·26-s + 14·27-s − 20·28-s − 3·29-s − 18·30-s + 3·31-s − 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.06·8-s − 9-s + 2.84·10-s − 2.30·12-s − 0.554·13-s + 4.00·14-s + 1.54·15-s + 3/4·16-s − 1.45·17-s + 2.12·18-s − 2.68·20-s + 2.18·21-s + 1.22·24-s + 1/5·25-s + 1.17·26-s + 2.69·27-s − 3.77·28-s − 0.557·29-s − 3.28·30-s + 0.538·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.528003\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8281,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad7$C_2$ \( 1 + 5 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ad_bi
37$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.37.a_bl
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cq
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2^2$ \( 1 + 15 T + 122 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_es
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ct
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_cw
73$C_2^2$ \( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.73.p_fs
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_acc
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.83.a_afy
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_fh
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.j_eu
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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

−13.63904160802417821624663628691, −13.16371348442870399093902062507, −12.42643715581397599566395255420, −11.93305911788362475343826251061, −11.45890042196068561769156560531, −11.21711552819622516178340218458, −10.25644898687776839236866277489, −10.12907394878540675614875167831, −9.395539239652842239684726576896, −8.791491402927148758584956909343, −8.434338348736293880930741680040, −7.88654935467074101815542443262, −6.78328450407209927392109956700, −6.76391275547223292903500878979, −5.86769882254247262044039795784, −4.99121960582268942783276276215, −3.72903192561893816413158387599, −2.84908049331121638651684860775, 0, 0, 2.84908049331121638651684860775, 3.72903192561893816413158387599, 4.99121960582268942783276276215, 5.86769882254247262044039795784, 6.76391275547223292903500878979, 6.78328450407209927392109956700, 7.88654935467074101815542443262, 8.434338348736293880930741680040, 8.791491402927148758584956909343, 9.395539239652842239684726576896, 10.12907394878540675614875167831, 10.25644898687776839236866277489, 11.21711552819622516178340218458, 11.45890042196068561769156560531, 11.93305911788362475343826251061, 12.42643715581397599566395255420, 13.16371348442870399093902062507, 13.63904160802417821624663628691

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