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Properties

Label 4-416e2-1.1-c1e2-0-25
Degree $4$
Conductor $173056$
Sign $-1$
Analytic cond. $11.0342$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 5·9-s + 2·13-s − 6·17-s − 7·25-s + 4·29-s + 10·37-s − 24·41-s − 10·45-s − 5·49-s + 8·53-s − 8·61-s + 4·65-s − 4·73-s + 16·81-s − 12·85-s + 4·89-s + 20·97-s + 8·101-s − 38·109-s − 12·113-s − 10·117-s − 18·121-s − 26·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 5/3·9-s + 0.554·13-s − 1.45·17-s − 7/5·25-s + 0.742·29-s + 1.64·37-s − 3.74·41-s − 1.49·45-s − 5/7·49-s + 1.09·53-s − 1.02·61-s + 0.496·65-s − 0.468·73-s + 16/9·81-s − 1.30·85-s + 0.423·89-s + 2.03·97-s + 0.796·101-s − 3.63·109-s − 1.12·113-s − 0.924·117-s − 1.63·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(173056\)    =    \(2^{10} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(11.0342\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 173056,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.a_bl
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.039528016718973273192662402272, −8.552472099700084871980246984615, −8.052297265441174562544922743090, −7.71262456178893161528956236885, −6.65408062467448396251062225874, −6.48929045010611195636913139276, −6.08119338191936199705626075673, −5.41207186938958458805928008478, −5.12564549383277447468893363985, −4.31704488135266103326965750563, −3.64849442843249490400769039530, −2.91927671703962873131976341826, −2.34351544305748892558428024728, −1.62873087004997173572398631453, 0, 1.62873087004997173572398631453, 2.34351544305748892558428024728, 2.91927671703962873131976341826, 3.64849442843249490400769039530, 4.31704488135266103326965750563, 5.12564549383277447468893363985, 5.41207186938958458805928008478, 6.08119338191936199705626075673, 6.48929045010611195636913139276, 6.65408062467448396251062225874, 7.71262456178893161528956236885, 8.052297265441174562544922743090, 8.552472099700084871980246984615, 9.039528016718973273192662402272

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