↑

Properties

Label 4-49923-1.1-c1e2-0-1
Degree $4$
Conductor $49923$
Sign $-1$
Analytic cond. $3.18313$
Root an. cond. $1.33571$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·4-s − 4·7-s + 9-s + 4·12-s + 6·13-s + 12·16-s + 4·19-s + 4·21-s − 6·25-s − 27-s + 16·28-s − 10·31-s − 4·36-s + 16·37-s − 6·39-s − 2·43-s − 12·48-s − 2·49-s − 24·52-s − 4·57-s − 16·61-s − 4·63-s − 32·64-s − 30·67-s + 24·73-s + 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 2·4-s − 1.51·7-s + 1/3·9-s + 1.15·12-s + 1.66·13-s + 3·16-s + 0.917·19-s + 0.872·21-s − 6/5·25-s − 0.192·27-s + 3.02·28-s − 1.79·31-s − 2/3·36-s + 2.63·37-s − 0.960·39-s − 0.304·43-s − 1.73·48-s − 2/7·49-s − 3.32·52-s − 0.529·57-s − 2.04·61-s − 0.503·63-s − 4·64-s − 3.66·67-s + 2.80·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49923 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49923 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49923\)    =    \(3^{3} \cdot 43^{2}\)
Sign: $-1$
Analytic conductor: \(3.18313\)
Root analytic conductor: \(1.33571\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 49923,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( 1 + T \)
43$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.13.ag_bj
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.41.a_bh
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.67.be_nv
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.a_acc
73$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.73.ay_le
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.97.aw_md
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

−9.761758139217697478982108964917, −9.360930401498180975205138352635, −9.000644041338349206828923104341, −8.510537086333408831410494114245, −7.64365379837943969382466665403, −7.51672561268763905855633948060, −6.27273821245987508595068296869, −5.96338923144874103909303632728, −5.71816265250030609184960935219, −4.74387622427760402111791858890, −4.27563665262617109733543688482, −3.47776319336265678852062283368, −3.29623080699256347856604127938, −1.27597065440948837170642206895, 0, 1.27597065440948837170642206895, 3.29623080699256347856604127938, 3.47776319336265678852062283368, 4.27563665262617109733543688482, 4.74387622427760402111791858890, 5.71816265250030609184960935219, 5.96338923144874103909303632728, 6.27273821245987508595068296869, 7.51672561268763905855633948060, 7.64365379837943969382466665403, 8.510537086333408831410494114245, 9.000644041338349206828923104341, 9.360930401498180975205138352635, 9.761758139217697478982108964917

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