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Properties

Label 4-2312e2-1.1-c1e2-0-7
Degree $4$
Conductor $5345344$
Sign $1$
Analytic cond. $340.823$
Root an. cond. $4.29667$
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  − 6·9-s + 4·13-s + 8·19-s − 2·25-s − 8·43-s + 16·47-s − 6·49-s + 20·53-s + 24·59-s + 24·67-s + 27·81-s + 8·83-s − 4·89-s − 12·101-s − 16·103-s − 24·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯
L(s)  = 1  − 2·9-s + 1.10·13-s + 1.83·19-s − 2/5·25-s − 1.21·43-s + 2.33·47-s − 6/7·49-s + 2.74·53-s + 3.12·59-s + 2.93·67-s + 3·81-s + 0.878·83-s − 0.423·89-s − 1.19·101-s − 1.57·103-s − 2.21·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5345344\)    =    \(2^{6} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(340.823\)
Root analytic conductor: \(4.29667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5345344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.450051283\)
\(L(\frac12)\) \(\approx\) \(2.450051283\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.31.a_cc
37$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.37.a_co
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.61.a_ek
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−9.188422496244036675285023161079, −8.555365019787868614097738869246, −8.497414989851784072260795788639, −8.266994327019313964515464863028, −7.74809682337731867011443975559, −7.21226778019007651959982871674, −6.86908410979874442107037569351, −6.53427021683811340721899550693, −5.88790768442191170555969404321, −5.55825837797709445858675244332, −5.39758477840499312834510340574, −5.14594820422196006112091889697, −4.15500868742058610996620963069, −3.93028872770119165447975231645, −3.23145387132707216836473075285, −3.22295812629432437428857419278, −2.31340952033643556350808867688, −2.17816937944146410272551954773, −0.987550290727745191555206437739, −0.67014433514962636519329363162, 0.67014433514962636519329363162, 0.987550290727745191555206437739, 2.17816937944146410272551954773, 2.31340952033643556350808867688, 3.22295812629432437428857419278, 3.23145387132707216836473075285, 3.93028872770119165447975231645, 4.15500868742058610996620963069, 5.14594820422196006112091889697, 5.39758477840499312834510340574, 5.55825837797709445858675244332, 5.88790768442191170555969404321, 6.53427021683811340721899550693, 6.86908410979874442107037569351, 7.21226778019007651959982871674, 7.74809682337731867011443975559, 8.266994327019313964515464863028, 8.497414989851784072260795788639, 8.555365019787868614097738869246, 9.188422496244036675285023161079

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