↑

Properties

Label 4-416e2-1.1-c1e2-0-3
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 3·9-s − 4·11-s + 5·13-s − 4·15-s − 3·17-s + 2·19-s − 2·23-s − 7·25-s − 10·27-s − 5·29-s + 4·31-s + 8·33-s − 5·37-s − 10·39-s − 3·41-s + 4·43-s + 6·45-s + 12·47-s + 7·49-s + 6·51-s + 26·53-s − 8·55-s − 4·57-s − 12·59-s + 7·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s + 1.38·13-s − 1.03·15-s − 0.727·17-s + 0.458·19-s − 0.417·23-s − 7/5·25-s − 1.92·27-s − 0.928·29-s + 0.718·31-s + 1.39·33-s − 0.821·37-s − 1.60·39-s − 0.468·41-s + 0.609·43-s + 0.894·45-s + 1.75·47-s + 49-s + 0.840·51-s + 3.57·53-s − 1.07·55-s − 0.529·57-s − 1.56·59-s + 0.896·61-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

\(L(1)\) \(\approx\) \(1.126908886\)
\(L(\frac12)\) \(\approx\) \(1.126908886\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_at
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_ae
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.37.f_am
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.53.aba_kp
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_ez
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36303698360904672155054736461, −10.98322650556110006452846065851, −10.49968898232328623053280887750, −10.24188008281258316411059250208, −9.818620437023757232388838948827, −9.133719478554666529755097779734, −8.943373161667052862379020486968, −8.057165138715353796603730874785, −7.80572251307510414354091038004, −7.11200806205607986272802979645, −6.71005873146354384170307873527, −5.93636381038984198334203286701, −5.67612001872800744029860585719, −5.55272950821372477571782559729, −4.78933096749039358130717592127, −3.85092699556815660406124571196, −3.76521666680915787686574047205, −2.31192757469160141331754803335, −2.01982745446997270784132451189, −0.74079073947940448674303916649, 0.74079073947940448674303916649, 2.01982745446997270784132451189, 2.31192757469160141331754803335, 3.76521666680915787686574047205, 3.85092699556815660406124571196, 4.78933096749039358130717592127, 5.55272950821372477571782559729, 5.67612001872800744029860585719, 5.93636381038984198334203286701, 6.71005873146354384170307873527, 7.11200806205607986272802979645, 7.80572251307510414354091038004, 8.057165138715353796603730874785, 8.943373161667052862379020486968, 9.133719478554666529755097779734, 9.818620437023757232388838948827, 10.24188008281258316411059250208, 10.49968898232328623053280887750, 10.98322650556110006452846065851, 11.36303698360904672155054736461

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