↑

Properties

Label 4-2394e2-1.1-c1e2-0-19
Degree $4$
Conductor $5731236$
Sign $1$
Analytic cond. $365.428$
Root an. cond. $4.37220$
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-s + 5-s + 2·7-s − 8-s + 10-s + 4·11-s + 5·13-s + 2·14-s − 16-s − 19-s + 4·22-s + 23-s + 5·25-s + 5·26-s + 6·29-s + 8·31-s + 2·35-s − 8·37-s − 38-s − 40-s − 2·41-s + 8·43-s + 46-s + 3·49-s + 5·50-s − 2·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.38·13-s + 0.534·14-s − 1/4·16-s − 0.229·19-s + 0.852·22-s + 0.208·23-s + 25-s + 0.980·26-s + 1.11·29-s + 1.43·31-s + 0.338·35-s − 1.31·37-s − 0.162·38-s − 0.158·40-s − 0.312·41-s + 1.21·43-s + 0.147·46-s + 3/7·49-s + 0.707·50-s − 0.274·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5731236\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(365.428\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5731236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.991804859\)
\(L(\frac12)\) \(\approx\) \(5.991804859\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_abl
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_abx
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ak
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cz
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fy
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.83.ao_ih
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2^2$ \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_bv
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

−8.923366005081995561376678456108, −8.766049285255243972653725299581, −8.481044083804738850281600738480, −8.225152873479505353024397754354, −7.44152417481062794098920154438, −7.28619362939549886473334309132, −6.51139073944148489943379009259, −6.46720097529748095166023875460, −6.09295100444871272002744588399, −5.68158016670952224657124663370, −5.08082397176612269558228985475, −4.79265747391555046582614016045, −4.35704175546473627047984511318, −4.06467640964997724175119809354, −3.32077157908290356468073734633, −3.23782375523766038743011811500, −2.45930340983725361587408871227, −1.87308658470553931678920740379, −1.23805533134719575055397976955, −0.859702679339627593443603714311, 0.859702679339627593443603714311, 1.23805533134719575055397976955, 1.87308658470553931678920740379, 2.45930340983725361587408871227, 3.23782375523766038743011811500, 3.32077157908290356468073734633, 4.06467640964997724175119809354, 4.35704175546473627047984511318, 4.79265747391555046582614016045, 5.08082397176612269558228985475, 5.68158016670952224657124663370, 6.09295100444871272002744588399, 6.46720097529748095166023875460, 6.51139073944148489943379009259, 7.28619362939549886473334309132, 7.44152417481062794098920154438, 8.225152873479505353024397754354, 8.481044083804738850281600738480, 8.766049285255243972653725299581, 8.923366005081995561376678456108

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