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Properties

Label 4-416e2-1.1-c1e2-0-27
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 21·9-s + 4·11-s − 6·17-s − 12·19-s − 9·25-s + 54·27-s + 24·33-s + 10·43-s − 13·49-s − 36·51-s − 72·57-s + 20·59-s + 4·67-s − 20·73-s − 54·75-s + 108·81-s + 12·89-s + 28·97-s + 84·99-s + 8·107-s + 4·113-s − 10·121-s + 127-s + 60·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s + 1.20·11-s − 1.45·17-s − 2.75·19-s − 9/5·25-s + 10.3·27-s + 4.17·33-s + 1.52·43-s − 1.85·49-s − 5.04·51-s − 9.53·57-s + 2.60·59-s + 0.488·67-s − 2.34·73-s − 6.23·75-s + 12·81-s + 1.27·89-s + 2.84·97-s + 8.44·99-s + 0.773·107-s + 0.376·113-s − 0.909·121-s + 0.0887·127-s + 5.28·129-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: \(0\)
Selberg data: \((4,\ 173056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.544989277\)
\(L(\frac12)\) \(\approx\) \(5.544989277\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.3.ag_p
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.47.a_acx
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.59.au_ik
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.a_en
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−9.175534225967753404519516979495, −8.567359956354704438407736576121, −8.542273196799616534200427297006, −7.938211248845934878732993890823, −7.53566579558648020530867420842, −6.92880645805086748841087984321, −6.53148501748885385532746785562, −5.98494658430119741237026147495, −4.43437896316507372499242030963, −4.42326331313024154570777237165, −3.79033676077549186455122166647, −3.47529908355398035375426875224, −2.47488980077832639227670038418, −2.17545984222807868215828295765, −1.74404149014246056815546005306, 1.74404149014246056815546005306, 2.17545984222807868215828295765, 2.47488980077832639227670038418, 3.47529908355398035375426875224, 3.79033676077549186455122166647, 4.42326331313024154570777237165, 4.43437896316507372499242030963, 5.98494658430119741237026147495, 6.53148501748885385532746785562, 6.92880645805086748841087984321, 7.53566579558648020530867420842, 7.938211248845934878732993890823, 8.542273196799616534200427297006, 8.567359956354704438407736576121, 9.175534225967753404519516979495

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